Kinematics - IB Math AA Question Bank

Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

A particle moves in a straight line. Its acceleration at time $t$ seconds is

a(t)=6t-6

At $t=0$ , its velocity is $8\text{ m s}^{-1}$ and its displacement from the origin is $0\text{ m}$ .

A

Find $v(t)$ , the velocity of the particle at time $t$ .

[2]

B

Find the time at which the speed of the particle is least.

[2]

C

Find the displacement of the particle when $t=2$ .

[1]

Question 2

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A particle moves in a straight line. Its displacement from a fixed origin at time $t$ seconds, $0\le t\le 5$ , is given by

s(t)=t^3-6t^2+9t

Here, $s$ is measured in metres.

A

Find the velocity of the particle and the times at which it is at rest.

[2]

B

State the intervals during which the particle is moving in the positive direction.

[1]

C

Find the total distance travelled by the particle in the first $5$ seconds.

[3]

Question 3

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A particle moves along a straight line with acceleration

a(t)=4-2t

where $t$ is measured in seconds. Initially, the particle has velocity $-1\text{ m s}^{-1}$ and displacement $5\text{ m}$ from the origin.

A

Find an expression for the velocity $v(t)$ .

[2]

B

Find the times in the interval $0\le t\le 4$ when the particle is at rest.

[2]

C

Determine whether the displacement has a local maximum or a local minimum at each of these times.

[1]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Two particles, $P$ and $Q$ , move along the same straight line. Their displacements from a fixed origin at time $t$ seconds are

s_P(t)=t^2-4t+1,\qquad s_Q(t)=7-2t

A

Find the time $t>0$ at which the particles meet.

[2]

B

Find the velocity of each particle at the time they meet.

[2]

C

State whether the particles are moving in the same direction when they meet. Give a reason.

[1]

Question 5

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line. Its displacement from a fixed origin after $t$ seconds is

s(t)=2t^3-15t^2+24t+5

where $0\leq t\leq 5$ and $s$ is measured in metres.

A

Find expressions for the velocity and acceleration of the particle at time $t$ .

[2]

B

Find the times when the particle is instantaneously at rest.

[2]

C

Find the total distance travelled by the particle during the first $5$ seconds.

[2]

Question 6

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The acceleration of a particle moving along a straight line is

a(t)=1.2t-4.8

where $0\leq t\leq 6$ . Initially, the particle is at the origin and has velocity $3\ \text{m s}^{-1}$ .

A

Find an expression for the velocity $v(t)$ .

[2]

B

Find the time in the interval when the particle is instantaneously at rest.

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=6$ .

[2]

Question 7

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The velocity-time graph of a particle moving along a straight line consists of straight line segments joining five labelled points, as shown.

Piecewise-linear velocity-time graph for a particle moving along a straight line.

A

Determine the acceleration of the particle for $2\leq t\leq 5$ .

[1]

B

Find the times after $t=0$ when the particle is instantaneously at rest.

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=9$ .

[3]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line with velocity

v(t)=0.5t^2-3t+2

where $0\leq t\leq 7$ and $t$ is measured in seconds.

A

Find the acceleration of the particle when $t=4$ .

[1]

B

Find the times when the particle is instantaneously at rest.

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=7$ .

[3]

Question 9

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The velocity of a particle moving along a straight line is given by

v(t)=\sin t+\cos t

for $0\le t\le \pi$ , where $t$ is measured in seconds and $v(t)$ is measured in $\text{m s}^{-1}$ . The particle is initially at the origin.

A

Find the acceleration of the particle at $t=\frac{\pi}{2}$ .

[2]

B

Find the displacement of the particle from $t=0$ to $t=\pi$ .

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=\pi$ .

[2]

Question 10

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The velocity $v\text{ m s}^{-1}$ of a particle moving in a straight line is defined by

v(t)= \begin{cases} 4-t, & 0\le t\le 6,\\ t-8, & 6<t\le 10. \end{cases}

A

Find the displacement of the particle from $t=0$ to $t=10$ .

[2]

B

Find the total distance travelled by the particle from $t=0$ to $t=10$ .

[4]

Question 11

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in the plane. Its position vector at time $t$ seconds, $0\le t\le 4$ , is

\mathbf r(t)=(t^2-4t)\mathbf i+(2t-t^2)\mathbf j

The position is measured in metres.

A

Find the velocity vector of the particle at time $t$ .

[2]

B

Find the time at which the speed of the particle is least.

[2]

C

Find the least speed of the particle.

[2]

Question 12

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in the plane with acceleration

\mathbf a=-2\mathbf j

At $t=0$ , its velocity is $3\mathbf i+4\mathbf j$ and its position vector is $\mathbf j$ .

A

Find the position vector $\mathbf r(t)$ of the particle.

[3]

B

Show that the path of the particle satisfies

y=1+\frac{4x}{3}-\frac{x^2}{9}

[2]

C

Find the maximum value of $y$ .

[1]

Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in the plane with position vector

\mathbf r(t)=e^t\mathbf i+e^{-t}\mathbf j

where $t\in\mathbb R$ .

A

Find the speed of the particle at time $t$ .

[2]

B

Determine the time at which the speed is least.

[2]

C

Show that, at this time, the velocity and acceleration vectors are perpendicular.

[1]

Question 14

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A particle moves in a straight line with velocity

v(t)=4\sin(0.8t)-1

where $0\leq t\leq 8$ , $t$ is measured in seconds and $v$ in $\text{m s}^{-1}$ .

A

Find the times in the interval when the particle is instantaneously at rest.

[2]

B

Find the displacement of the particle from $t=0$ to $t=8$ .

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=8$ .

[3]

Question 15

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The displacement of a particle from a fixed origin is modelled by

s(t)=8\ln(t+1)-t^2+3

where $0\leq t\leq 6$ , $t$ is measured in seconds and $s$ in metres.

A

Find the velocity of the particle at time $t$ .

[1]

B

Find the time when the particle changes direction.

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=6$ .

[3]

Question 16

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line. Let $s$ be the numerical value of its displacement in metres and let $v$ be the numerical value of its velocity in $\text{m s}^{-1}$ . They are related by

v(s)=\frac{s}{s+1}, \qquad s\geq 0

The acceleration $a$ is expressed by its numerical value in $\text{m s}^{-2}$ .

A

Show that the acceleration is $a(s)=\dfrac{s}{(s+1)^3}$ .

[3]

B

Find the displacement at which the acceleration is greatest.

[2]

C

Find the values of $s$ for which the acceleration is $0.100\ \text{m s}^{-2}$ .

[2]

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The position vector of a particle at time $t$ seconds is

\mathbf r(t)=(t^3-6t)\mathbf i+4e^{-t/2}\mathbf j

where $0\leq t\leq 3$ and distances are measured in metres.

A

Find the velocity vector and the acceleration vector of the particle at time $t$ .

[2]

B

Find the speed of the particle when $t=2$ .

[2]

C

Determine the time at which the velocity vector is perpendicular to the acceleration vector.

[2]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line with velocity

v(t)=0.2(t-0.5)(t-3)(t-6)

where $0\leq t\leq 8$ , $t$ is measured in seconds and $v$ in $\text{m s}^{-1}$ .

A

Find the times when the particle is instantaneously at rest.

[2]

B

Calculate the displacement of the particle from $t=0$ to $t=8$ .

[2]

C

Calculate the total distance travelled by the particle from $t=0$ to $t=8$ .

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line. Its position coordinate at time $t$ seconds is

x(t)=t^2-4t+3

where $0\leq t\leq 3$ and distances are measured in metres.

A

Find the velocity of the particle at time $t$ .

[2]

B

Find the speed and the magnitude of the acceleration of the particle when $t=1$ .

[2]

C

Find the times, with $0<t\leq 3$ , when the particle is at the origin.

[2]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A particle moves in a plane with position vector

\mathbf r(t)=3\sin t\,\mathbf i+4\cos t\,\mathbf j

where $0\leq t\leq \pi$ , $t$ is measured in seconds and distances are measured in metres. This question intentionally extends one-dimensional kinematics to planar motion; numerical integration may be used where necessary.

A

Find the velocity vector and acceleration vector of the particle at time $t$ .

[2]

B

Find the maximum speed of the particle and the time at which it occurs.

[2]

C

Calculate the total distance travelled by the particle from $t=0$ to $t=\pi$ .

[2]

Question 21

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A particle $A$ moves in a straight line. Its displacement from a fixed origin at time $t$ seconds is given by

s_A(t)=t^3-9t^2+24t

where $0\le t\le 6$ and $s_A$ is measured in metres.

A

I.

Find an expression for the velocity of particle $A$ at time $t$ .

[2]

II.

Find the times at which particle $A$ is instantaneously at rest, and state the direction of motion in each of the intervals determined by these times.

[3]

B

I.

Find the displacement of particle $A$ from the origin when $t=2$ , $t=4$ and $t=6$ .

[2]

II.

Hence find the total distance travelled by particle $A$ during the first $6$ seconds.

[3]

C

A second particle $B$ moves along the same line with velocity $v_B(t)=10-t$ , for $0\le t\le 10$ . Particle $B$ starts from the origin. Find the first time at which the distance travelled by particle $B$ is equal to the total distance travelled by particle $A$ during the first $6$ seconds. If you did not obtain an answer to part (b)(ii), use $44\text{ m}$ .

[2]

Question 22

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A particle moves in a straight line with velocity

v(t)=2\cos t-1

where $0\le t\le 2\pi$ . Initially, the particle is $1\text{ m}$ from the origin in the positive direction.

A

(a)

I.

Find the acceleration of the particle at time $t$ .

[1]

II.

Find the times at which the particle is instantaneously at rest.

[3]

B

(b)

I.

Find an expression for the displacement $s(t)$ of the particle from the origin.

[2]

II.

Find the displacement of the particle from $t=0$ to $t=2\pi$ .

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=2\pi$ . If you did not obtain the rest times in part (a)(ii), use $t=\frac{\pi}{3}$ and $t=\frac{5\pi}{3}$ .

[3]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The velocity-time graph of a particle moving in a straight line consists of straight line segments joining the points shown. The displacement of the particle from the origin at $t=0$ is $-5\text{ m}$ .

A

I.

Find the acceleration of the particle for $0\le t\le 2$ .

[2]

II.

Find all times at which the particle is instantaneously at rest.

[2]

B

I.

Find the displacement of the particle from $t=0$ to $t=10$ .

[2]

II.

Find the displacement of the particle from the origin at $t=10$ .

[2]

C

I.

Find the total distance travelled by the particle from $t=0$ to $t=10$ .

[2]

II.

Find the first time at which the particle passes through the origin.

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A particle moves in a straight line with velocity

v(t)=t^2-4t+3

where $0\le t\le 5$ . The particle starts at the origin. Here, $t$ is measured in seconds and $v(t)$ in metres per second.

A

I.

Find the times at which the particle is instantaneously at rest.

[2]

II.

Find an expression for the displacement $s(t)$ of the particle from the origin.

[2]

III.

Show that the particle returns to the origin when $t=3$ .

[1]

B

I.

State the intervals during which the particle moves in the positive direction.

[1]

II.

Find the total distance travelled by the particle from $t=0$ to $t=5$ . If you did not obtain the displacement function in part (a)(ii), use $s(t)=\frac{t^3}{3}-2t^2+3t$ .

[2]

C

Find the average velocity and the average speed of the particle over the interval $0\le t\le 5$ .

[2]

Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in three-dimensional space with position vector

\mathbf r(t)=3\cos t\,\mathbf i+3\sin t\,\mathbf j+4t\,\mathbf k

where $t$ is measured in seconds and distances are measured in metres. Here $\mathbf i$ , $\mathbf j$ , and $\mathbf k$ are mutually perpendicular unit vectors along the positive $x$ -, $y$ -, and $z$ -axes, respectively.

A

I.

Find the velocity vector and acceleration vector of the particle at time $t$ .

[2]

II.

Show that the velocity and acceleration vectors are perpendicular for all $t$ in this interval.

[2]

B

I.

Find the speed of the particle.

[2]

II.

Find the total distance travelled by the particle for $0\le t\le \pi$ .

[1]

C

Find the angle between the velocity vector and the positive $z$ -axis.

[3]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in a straight line. Its velocity $v\text{ m s}^{-1}$ at time $t$ seconds satisfies

\frac{dv}{dt}=3-v

Initially, $v=1$ and the particle is at the origin.

A

By solving the differential equation, show that

v(t)=3-2e^{-t}

[3]

B

Find the time at which the velocity is $2\text{ m s}^{-1}$ .

[1]

C

Find the displacement of the particle at the time found in part (b).

[2]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A particle moves in a straight line. Its acceleration $a\text{ m s}^{-2}$ is related to its displacement $s\text{ m}$ from the origin by

a=4s

When $s=0$ , the particle has velocity $3\text{ m s}^{-1}$ . The particle continues to move in the positive direction.

A

Show that

v^2=4s^2+9

[3]

B

Find the velocity of the particle when $s=2$ .

[1]

C

Find the acceleration of the particle when $s=2$ .

[1]

Question 28

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Two particles, $P$ and $Q$ , move in three-dimensional space. Their position vectors at time $t$ seconds, $t\ge 0$ , are

\mathbf r_P=(1,0,2)+t(2,-1,1)

\mathbf r_Q=(5,-2,0)+t(0,1,2)

A

Find the position vector of $P$ relative to $Q$ at time $t$ .

[2]

B

Find the time at which the distance between the particles is least.

[3]

C

Find the least distance between the particles.

[1]

Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A lift moves vertically along a straight shaft. Its velocity, in $\text{m s}^{-1}$ , at time $t$ seconds is modelled by

v(t)=1.8\sin\left(\frac{\pi t}{6}\right)-0.3,\qquad 0\le t\le 12

Positive velocity is upwards. At $t=0$ , the lift is $2\text{ m}$ above the ground floor.

Velocity-time graph of a lift over 12 s.

A

I.

Find the times at which the lift is instantaneously at rest.

[3]

II.

State the time interval during which the lift is moving upwards.

[2]

B

I.

Find the height of the lift above the ground floor when it is at its maximum height.

[3]

II.

Find the displacement of the lift from $t=0$ to $t=12$ .

[1]

C

Calculate the total distance travelled by the lift during the $12$ seconds.

[3]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A particle moves along a straight line. Its displacement from a fixed origin at time $t$ seconds is

s(t)=3+6t-2.5t^2+0.2t^3,\qquad 0\le t\le 8

where $s$ is measured in metres.

A

I.

Find an expression for the velocity of the particle.

[2]

II.

Find the times at which the particle changes direction.

[3]

B

I.

Find the acceleration of the particle at time $t$ .

[1]

II.

Find the velocity of the particle when its acceleration is zero.

[2]

C

Calculate the total distance travelled by the particle during the interval $0\le t\le 8$ .

[3]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The velocity-time graph of a particle moving in a straight line consists of straight line segments joining the points shown. The velocity is measured in $\text{m s}^{-1}$ and time is measured in seconds.

A

I.

The graph passes through $(0,2)$ , $(3,5)$ , $(6,-1)$ , $(9,-1)$ and $(12,4)$ . Find the acceleration for $3<t<6$ .

[2]

II.

Find the times when the particle is instantaneously at rest.

[2]

B

I.

Calculate the displacement of the particle from $t=0$ to $t=12$ .

[2]

II.

Calculate the total distance travelled from $t=0$ to $t=12$ .

[2]

C

Given that the particle starts at the origin, find the greatest displacement of the particle from the origin during the interval $0\le t\le 12$ .

[2]

Question 32

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A projectile is launched from a point $1\ \text{m}$ above horizontal ground. Its position vector at time $t$ seconds is

\mathbf r(t)=8t\mathbf i+(1+12t-4.9t^2)\mathbf j

where distances are measured in metres and $\mathbf j$ is vertically upwards.

A

I.

Find the velocity vector of the projectile at time $t$ .

[2]

II.

Find the acceleration vector of the projectile.

[2]

B

I.

Find the maximum height reached by the projectile.

[2]

II.

Find the horizontal distance travelled when the projectile first reaches the ground.

[1]

C

Determine the speed and the direction of motion of the projectile at the instant it reaches the ground. Give the direction as an angle below the horizontal.

[3]

Question 33

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A particle moves in three-dimensional space. Use the HL vector methods for three-dimensional kinematics in this question. Its position vector at time $t$ seconds is

\mathbf r(t)=t^2\mathbf i+\sin t\,\mathbf j+e^{-t}\mathbf k

where $0\leq t\leq 4$ and distances are measured in metres.

A

Find the speed of the particle when $t=1$ .

[2]

B

Find the minimum distance of the particle from the origin during the interval.

[3]

C

Determine whether the particle is moving towards or away from the origin when $t=1$ . Justify your answer.

[2]

Question 34

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Two delivery carts $P$ and $Q$ move along the same straight track. At time $t$ seconds, $P$ has velocity $v_P(t)=2+3e^{-t/2}$ and starts from the origin. Cart $Q$ starts $18\ \text{m}$ ahead of $P$ and moves with constant velocity $1\ \text{m s}^{-1}$ .

Displacement-time graph for carts P and Q on a straight track.

A

I.

Find the displacement of $P$ from the origin at time $t$ .

[2]

II.

Write down the displacement of $Q$ from the origin at time $t$ .

[2]

B

Determine the time at which $P$ catches $Q$ .

[3]

C

Justify that the carts meet only once for $t>0$ .

[3]

Question 35

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

A particle moves along a straight line. Its acceleration at time $t$ seconds is

a(t)=4\cos(2t)

where $0\le t\le \pi$ . Initially its velocity is $-1\text{ m s}^{-1}$ and it is at the origin.

A

I.

Find an expression for the velocity $v(t)$ of the particle.

[2]

II.

Find the times at which the particle is instantaneously at rest.

[3]

B

I.

Find an expression for the displacement $s(t)$ of the particle from the origin.

[2]

II.

Find the displacement of the particle from $t=0$ to $t=\pi$ .

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=\pi$ . If you did not obtain the rest times in part (a)(ii), use $t=\frac{\pi}{12}$ and $t=\frac{5\pi}{12}$ .

[4]

Question 36

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

A particle moves along a straight line with velocity

v(t)=e^{-t}-\frac14

where $0\le t\le \ln 16$ . The particle starts at the origin.

A

Part (a)

I.

Find the acceleration of the particle at time $t$ .

[1]

II.

Find the time at which the particle is instantaneously at rest.

[2]

III.

State whether this time gives a maximum or a minimum displacement. Give a reason.

[1]

B

Part (b)

I.

Find an expression for the displacement $s(t)$ of the particle from the origin.

[2]

II.

Find the displacement of the particle from $t=0$ to $t=\ln 16$ .

[1]

C

Find the total distance travelled by the particle from $t=0$ to $t=\ln16$ . If you did not obtain the time in part (a)(ii), use $t=\ln4$ .

[3]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Two particles $P$ and $Q$ move along a straight line. Taking a fixed positive direction along the line, the signed relative displacement of $P$ from $Q$ at time $t$ is

s(t)=\big((t-1)^2+1\big)\ \text{m}

where $t$ is the numerical value of the time in seconds and $0\le t\le 3$ .

A

I.

Find the relative velocity $v(t)$ and the relative acceleration $a(t)$ .

[2]

II.

Show that the square of the distance between the particles is $u^4+2u^2+1\ \text{m}^2$ , where $u=t-1$ .

[2]

B

I.

Determine the time at which the distance between the particles is least.

[2]

II.

Find the least distance between the particles.

[2]

C

Determine whether the particles are moving towards each other or away from each other when $t=2$ . Justify your answer.

[3]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A particle moves in a straight line. Its velocity $v\text{ m s}^{-1}$ satisfies the differential equation

\frac{dv}{dt}=2-2v

At $t=0$ , the particle is at the origin and has velocity $5\text{ m s}^{-1}$ .

A

I.

Show that $v(t)=1+4e^{-2t}$ .

[3]

II.

Find the acceleration of the particle at time $t$ .

[1]

B

I.

Find the time at which the acceleration is $-1\text{ m s}^{-2}$ .

[2]

II.

Explain why the particle never changes direction.

[1]

C

Find the displacement of the particle from the origin at the time found in part (b)(i). If you did not obtain this time, use $t=\frac32\ln2$ .

[3]

Question 39

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Two small robots, $P$ and $Q$ , move along the same straight track. At $t=0$ , robot $P$ is at the origin and robot $Q$ is $18\ \text{m}$ ahead of $P$ . Their velocities, in $\text{m s}^{-1}$ , are

v_P(t)=4+2\sin(0.5t),\qquad v_Q(t)=7-0.4t

for $0\le t\le 20$ .

A

AI.

Find expressions for the displacement of each robot from the origin at time $t$ .

[3]

AII.

Find the time, after $t=0$ , when the robots meet.

[2]

B

At the instant when the robots meet, determine whether they are moving in the same direction. Give a reason.

[3]

C

CI.

Find the position of the robots when they meet.

[2]

CII.

Calculate the total distance travelled by robot $Q$ before the robots meet.

[2]

Question 40

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A particle moves along a straight line. Its acceleration at time $t$ seconds is

a(t)=t-5,\qquad 0\le t\le 10

Initially, the particle is $2\ \text{m}$ to the negative side of the origin and has velocity $8\ \text{m s}^{-1}$ .

A

I.

Find an expression for the velocity $v(t)$ .

[2]

II.

Find an expression for the displacement $s(t)$ from the origin.

[3]

B

I.

Find the times at which the particle is instantaneously at rest.

[2]

II.

Determine the minimum velocity of the particle during the interval.

[1]

C

I.

Find the times in $0\le t\le 10$ when the particle is at the origin.

[2]

II.

Calculate the total distance travelled by the particle from $t=0$ to $t=10$ .

[2]

Question 41

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The displacement, in metres, of a sliding door from its closed position is modelled by

s(t)=12(1-e^{-0.3t})-0.8t,\qquad 0\le t\le 15

where $t$ is measured in seconds. Positive displacement means that the door is opening.

A

I.

Find the velocity of the door at time $t$ .

[2]

II.

Find the acceleration of the door at time $t$ .

[2]

B

I.

Find the time at which the door is furthest from the closed position.

[2]

II.

Find the maximum displacement of the door from the closed position.

[2]

C

I.

Find the times when the speed of the door is $0.5\ \text{m s}^{-1}$ .

[2]

II.

Calculate the total distance travelled by the door from $t=0$ to $t=15$ .

[1]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A particle moves along a straight line. Its velocity $v\ \text{m s}^{-1}$ satisfies the differential equation

\frac{dv}{dt}=0.6(5-v)

At $t=0$ , $v=-2$ and the displacement from the origin is $0\ \text{m}$ .

A

I.

Show that $v(t)=5-7e^{-0.6t}$ .

[3]

II.

Find the time when the particle first comes to rest.

[2]

B

I.

Find an expression for the displacement $s(t)$ from the origin.

[2]

II.

Find the displacement of the particle at $t=8$ .

[1]

C

I.

Calculate the total distance travelled by the particle during the first $8$ seconds.

[2]

II.

Find the time at which the particle is $20\ \text{m}$ from the origin in the positive direction.

[1]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two drones, $A$ and $B$ , move in three-dimensional space. Their position vectors, in metres, at time $t$ seconds are

\mathbf r_A=(1,2,-1)+t(4,-1,2),\qquad \mathbf r_B=(13,-4,5)+t(-1,2,0)

A

I.

Find the position vector of drone $A$ relative to drone $B$ at time $t$ .

[2]

II.

Find the relative velocity of drone $A$ with respect to drone $B$ .

[2]

B

I.

Find the time at which the drones are closest together.

[2]

II.

Find the minimum distance between the drones.

[2]

C

Find the times when the drones are exactly $10\ \text{m}$ apart.

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves along a straight line with velocity $v(t)=(t-1)(t-4)$ , for $0\le t\le 6$ , where $t$ is measured in seconds and $v$ in $\text{m s}^{-1}$ .

Velocity-time curve for a quadratic particle motion on 0 to 6 s.

A

I.

Find the times at which the particle is instantaneously at rest.

[2]

II.

State the intervals on which the particle is moving in the positive direction.

[2]

B

Find the displacement of the particle from $t=0$ to $t=6$ .

[2]

C

Find the total distance travelled by the particle from $t=0$ to $t=6$ .

[3]

D

The model is generalized by defining the numerical value of the velocity (in $\text{m s}^{-1}$ ) as $v_p(t)=(t-p)(t-4p)$ for $0\le t\le 6p$ , where $t$ denotes the numerical value of time in seconds and $p>0$ is a dimensionless parameter. Show that the total distance travelled is $15p^3\ \text{m}$ .

[3]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves along a straight line. Its acceleration is $a(t)=\dfrac{6}{(t+1)^3}-1$ , for $0\le t\le 5$ . Initially the particle has velocity $0\ \text{m s}^{-1}$ and displacement $2\ \text{m}$ .

Acceleration as a function of time on 0 to 5 seconds, decreasing from positive to negative.

A

I.

Find $v(t)$ , the velocity at time $t$ .

[3]

II.

Find $s(t)$ , the displacement at time $t$ .

[2]

B

Find the time after $t=0$ at which the particle changes direction.

[3]

C

Find the total distance travelled from $t=0$ to $t=5$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two drones move in three-dimensional space. Their position vectors, in metres, are $\mathbf r_P=(3,1,5)+t(2,-1,3)$ and $\mathbf r_Q=(13,-3,7)+t(-1,1,4)$ , where $t\ge 0$ is measured in seconds.

A schematic three-dimensional diagram showing two drones moving along straight-line paths, with their initial positions and velocity directions indicated.

A

I.

Find the position vector of $Q$ relative to $P$ at time $t$ .

[2]

II.

Write down the relative velocity of $Q$ with respect to $P$ .

[2]

B

Find the time at which the distance between the drones is least.

[3]

C

Find the least distance between the drones.

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A train starts from rest at a station and travels for $8$ seconds before stopping at the next signal. Its velocity is modelled by $v(t)=kt(8-t)$ , $0\le t\le 8$ , where $k>0$ . The distance between the station and the signal is $48\ \text{m}$ .

Velocity-time curve for the train journey.

A

I.

Find the value of $k$ .

[3]

II.

Using the velocity-time graph, determine the time at which the maximum velocity occurs and the maximum velocity of the train.

[2]

B

Find the distance travelled in the first $3$ seconds.

[2]

C

A similar journey of length $D$ metres over time $T$ seconds is modelled by $v(t)=kt(T-t)$ , $0\le t\le T$ . Show that the maximum velocity is $\dfrac{3D}{2T}$ .

[3]

D

For a journey of length $60\ \text{m}$ , determine the least possible value of $T$ if the maximum velocity must not exceed $8\ \text{m s}^{-1}$ .

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $0\le t\le 4$ , the velocity of a particle moving along a straight line is $v(t)=t^2-4t+3$ .

Quadratic velocity-time graph for a particle on 0≤t≤4.

A

I.

Find the acceleration of the particle at time $t$ .

[2]

II.

Find the times at which the particle is at rest.

[2]

B

Find the displacement from $t=0$ to $t=4$ .

[2]

C

Find the total distance travelled from $t=0$ to $t=4$ .

[2]

D

The model is changed to $v(t)=t^2-4t+c$ on the same interval. Find the value of $c$ for which the displacement from $t=0$ to $t=4$ is zero.

[2]

Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves in the plane with position vector $\mathbf r(t)=t\mathbf i+\ln(1+t)\mathbf j$ , for $0\le t\le 5$ , where $t$ is measured in seconds and distances are measured in metres. This question assesses the planar extension of one-dimensional motion; use vector differentiation for velocity and acceleration, and calculate distance travelled by integrating speed.

Trajectory of the particle on y = ln(1+x) for 0≤x≤5.

A

I.

Find the velocity vector at time $t$ .

[2]

II.

Find the speed at time $t$ .

[2]

B

Show that the speed is decreasing for $0\le t\le 5$ .

[3]

C

Calculate the total distance travelled by the particle from $t=0$ to $t=5$ .

[2]

D

Find the magnitude of the acceleration when $t=2$ .

[2]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves along a straight line with velocity $v(t)=e^{-t}(4-t)$ for all $t\ge 0$ . Initially, its displacement from the origin is $0$ .

Velocity of the particle as a function of time, with a zoomed curve for 3.5 ≤ t ≤ 6 so that negative velocities are clearly distinguishable from v = 0.

A

I.

Find the time at which the particle changes direction.

[2]

II.

Find the acceleration at time $t$ .

[2]

B

Find the displacement of the particle from $t=0$ to $t=6$ .

[2]

C

Find the total distance travelled from $t=0$ to $t=6$ .

[3]

D

State the limiting displacement as $t\to\infty$ .

[1]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A particle moves in the plane with position vector

\mathbf r(t)=(t^2-2t)\mathbf i+\left(\frac{t^3}{3}-2t\right)\mathbf j

where $0\le t\le 3$ and distances are measured in metres.

A

I.

Find the velocity vector and acceleration vector of the particle at time $t$ .

[2]

II.

Find the time at which the velocity is parallel to the $x$ -axis.

[2]

B

I.

Show that the square of the speed is $t^4-8t+8$ .

[2]

II.

Determine the time at which the speed is least.

[2]

C

I.

Find the least speed of the particle.

[2]

II.

Show that, at the time when the speed is least, the velocity and acceleration vectors are perpendicular. If you did not obtain the time in part (b)(ii), use $t=\sqrt[3]{2}$ .

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A particle moves in a straight line. Its acceleration $a\text{ m s}^{-2}$ is related to its displacement $s\text{ m}$ from the origin by

a=-s

When $t=0$ , the particle is at the origin and has velocity $2\text{ m s}^{-1}$ . Initially the particle moves in the positive direction.

A

I.

Using $a=v\frac{dv}{ds}$ , show that $v^2=4-s^2$ .

[3]

II.

Find the maximum displacement of the particle from the origin.

[1]

B

I.

For the initial motion in the positive direction, show that $s=2\sin t$ .

[3]

II.

Find the time at which the particle first reaches its maximum displacement.

[2]

C

The motion continues according to $s=2\sin t$ for $0\le t\le \pi$ . Find the total distance travelled by the particle during this interval.

[3]

Question 53

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A particle moves in a straight line. Its displacement from a fixed origin at time $t$ seconds is

s(t)=t^2-4t+|t-2|+3

where $0\le t\le 5$ and $s$ is measured in metres.

A

I.

Write $s(t)$ as a piecewise-defined function without using the modulus sign.

[2]

II.

Show that $s(t)$ is continuous but not differentiable at $t=2$ .

[2]

B

I.

Find the velocity of the particle for $0<t<2$ and for $2<t<5$ .

[2]

II.

Explain why the particle changes direction at $t=2$ even though it is not instantaneously at rest at that time.

[1]

C

I.

Find the displacement of the particle from $t=0$ to $t=5$ .

[1]

II.

Find the total distance travelled by the particle from $t=0$ to $t=5$ .

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A particle moves in a straight line in the positive direction. Its acceleration $a\ \text{m s}^{-2}$ is related to its displacement $s\ \text{m}$ from a fixed origin by

a=12-0.5s

When $s=0$ , the particle has velocity $6\ \text{m s}^{-1}$ .

A

I.

Show that $v^2=36+24s-0.5s^2$ .

[3]

II.

Find the velocity of the particle when $s=30$ .

[2]

B

I.

Find the displacement at which the speed is greatest.

[2]

II.

Find the greatest speed of the particle.

[2]

C

I.

Find the time taken for the particle to move from $s=0$ to $s=10$ .

[2]

II.

Find the time taken for the particle to reach its greatest speed.

[1]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A particle moves in the plane with position vector

\mathbf r(t)=e^{-0.2t}\cos t\mathbf i+e^{-0.2t}\sin t\mathbf j,\qquad t\ge0

where distances are measured in metres and $t$ is measured in seconds.

A

I.

Find the velocity vector $\mathbf v(t)$ .

[3]

II.

Show that the speed of the particle is $\sqrt{1.04}e^{-0.2t}$ .

[2]

B

I.

Find the distance travelled by the particle from $t=0$ to $t=10$ .

[2]

II.

Find the time at which the particle is $0.5\ \text{m}$ from the origin.

[1]

C

I.

Show that the angle between the position vector and the velocity vector is constant.

[3]

II.

Find this angle in degrees.

[1]

Question 56

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A particle moves along a straight line. Its velocity, in $\text{m s}^{-1}$ , is given by

v(t)=|t-3|-1,\qquad 0\le t\le 8

At $t=0$ , the particle is at the origin.

Velocity-time graph of v(t)=|t-3|-1 on 0≤t≤8.

A

I.

Write $v(t)$ as a piecewise function without using the absolute value sign.

[2]

II.

Find the times at which the particle is instantaneously at rest.

[2]

B

I.

Find the displacement of the particle from $t=0$ to $t=8$ .

[2]

II.

Calculate the total distance travelled from $t=0$ to $t=8$ .

[2]

C

I.

Find the acceleration of the particle for $t<3$ and for $t>3$ .

[2]

II.

Discuss whether the acceleration is defined at $t=3$ .

[1]

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves along a straight line with velocity $v(t)=3\sin t-1$ , for $0\le t\le 2\pi$ .

Velocity-time curve for v(t)=3 sin t - 1 on 0≤t≤2π, with the two zeros marked.

A

I.

Show that the particle is instantaneously at rest at $t=\alpha$ and $t=\pi-\alpha$ , where $\alpha=\arcsin\left(\frac13\right)$ .

[2]

II.

State the interval on which the particle moves in the positive direction.

[2]

B

Find the displacement of the particle from $t=0$ to $t=2\pi$ .

[2]

C

Show that the total distance travelled is $8\sqrt2+4\alpha$ metres.

[4]

D

Hence find the average speed of the particle over the interval $0\le t\le 2\pi$ .

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves along a straight line so that its acceleration is related to its displacement $s$ from the origin by $a=-9s$ . At $t=0$ , $s=2$ and $v=0$ . The particle initially moves towards the origin.

Phase diagram of velocity v against displacement s.

A

I.

Show that $v^2=36-9s^2$ .

[3]

II.

Find the velocity of the particle when $s=1$ .

[2]

B

Show that the displacement may be written as $s=2\cos(3t)$ .

[3]

C

Find the first time at which the particle passes through the origin.

[2]

D

Find the total distance travelled in one complete oscillation.

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A test vehicle starts and finishes at rest during a journey of length $200\ \text{m}$ . Its velocity is modelled by $v(t)=kt^2(10-t)^2$ , for $0\le t\le 10$ , where $k>0$ .

Symmetric vehicle velocity-time curve on 0 to 10 s.

A

I.

Find the value of $k$ .

[3]

II.

Find the maximum velocity of the vehicle.

[2]

B

A similar journey of length $D$ metres in $T$ seconds is modelled by $v(t)=kt^2(T-t)^2$ . Show that $v_{\max}=\dfrac{15D}{8T}$ .

[4]

C

For a journey of length $200\ \text{m}$ , find the least journey time $T$ if the maximum velocity must not exceed $30\ \text{m s}^{-1}$ .

[3]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A particle moves in the plane with position vector $\mathbf r(t)=e^{-t/2}\cos t\,\mathbf i+e^{-t/2}\sin t\,\mathbf j$ , for $t\ge 0$ , where the coordinates are measured in metres and $t$ is measured in seconds. Use planar vector kinematics for this question.

Decaying spiral path traced by the particle in the plane.

A

I.

Find the velocity vector of the particle.

[2]

II.

Show that the speed is $\frac{\sqrt5}{2}e^{-t/2}$ .

[2]

B

Find the total distance travelled during the first complete revolution.

[3]

C

Find the total distance travelled as $t\to\infty$ .

[3]