What kinematics is trying to do

Kinematics is the branch of mechanics that describes motion without dealing with the forces that cause it. That last phrase matters. In this topic we ask where is it?, how fast is it moving?, and how is that motion changing? The question of why the motion changes belongs to forces in A.2.

We describe a moving body quantitatively using position, velocity and acceleration. Position is a vector quantity that gives the location of a body relative to a chosen origin. To state a position properly, you need a coordinate system, a direction convention and units. Without those choices, “the ball is 3 m away” is not really a physics statement yet.

For one-dimensional motion, a single axis is enough. In two dimensions, you need two coordinates; in three dimensions, three coordinates. You can choose the axes, but once you've chosen them, stay consistent. A projectile problem, for example, is usually much easier with horizontal and vertical axes.

Scalars, vectors and signs

A scalar is a physical quantity with magnitude but no direction. Distance, speed, mass, time and energy are scalars. A vector is a physical quantity with both magnitude and direction. Displacement, velocity, acceleration and force are vectors.

A minus sign on its own does not prove that a quantity is a vector. Temperature change, electric charge and energy can be negative in particular contexts while still being scalars. With vectors in one dimension, though, a sign is often used to show direction: if right is chosen as positive, then a velocity of −2 m s⁻¹ means 2 m s⁻¹ to the left.

Describing motion qualitatively and quantitatively

A qualitative description uses words such as “speeding up”, “slowing down”, “moving back towards the start” or “travelling in a curved path”. A quantitative description adds numbers, units and directions. Good physics usually needs both. The numbers give the size of the effect; the words keep the motion being described clear.

The main modelling idea here is simple: position changes with time. If we know the pattern of change, we can predict the future position. That prediction is only as good as the assumptions in the model: constant acceleration, negligible fluid resistance, locally flat Earth and treating the object as a point are all assumptions we will use carefully.

Velocity

Velocity is a vector quantity: it tells you how quickly position changes with time. In one dimension,

v = Δx/Δt, where v is velocity (m s⁻¹), Δx is the change in position along the chosen axis (m), and Δt is the time interval (s).

The magnitude of velocity is speed only when the motion follows the path direction being considered. Once the direction changes, speed and velocity need to be handled separately.

Speed is a scalar quantity that tells you how fast distance is travelled. For a steady journey, speed = distance travelled ÷ time taken. The unit m s⁻¹ literally means “metres per second”, so a speed of 4 m s⁻¹ means 4 m of path length every second.

Acceleration

Acceleration is a vector quantity: it tells you how quickly velocity changes with time:

a = Δv/Δt, where a is acceleration (m s⁻²).

Read the unit m s⁻² carefully. An acceleration of 3 m s⁻² means the velocity changes by 3 m s⁻¹ every second, in the direction of the acceleration. Acceleration is not the same as “going fast”. A train moving at constant high speed has zero acceleration; a ball at the top of its flight has zero vertical velocity at that instant but still has downward acceleration.

Direction of acceleration

Acceleration points in the direction of the change in velocity, not always in the direction of motion. If an object moves to the right but slows down, its acceleration is to the left. If it moves to the left but slows down, its acceleration is to the right. This sign logic is a small thing, but it saves many answers.

Near Earth’s surface, freely falling bodies are often modelled using gravitational acceleration, an acceleration caused by Earth’s gravitational field. We usually use g = 9.81 m s⁻² or 9.8 m s⁻², where g is the magnitude of the acceleration due to gravity close to Earth’s surface (m s⁻²). In equations, its sign depends on the axis direction you choose.

Change in position

Displacement is a vector quantity that gives the change in position of a body from its initial position to its final position. It says nothing about the route taken. A student might walk around several streets to reach school, while another could somehow travel straight from home to school; if they start and finish at the same places, their displacement is the same.

In one dimension, the sign of displacement depends on the direction convention. If upwards is positive, a ball that starts at 1.2 m above the ground and later reaches 0.9 m above the ground has a displacement of −0.3 m. The minus sign shows that the change in position is downward.

Components of displacement

A vector can be resolved into perpendicular components. On a slope, for example, the displacement along the slope can be split into a horizontal component and a vertical component. This isn't just mathematical decoration; it is exactly the habit needed later for projectile motion, where horizontal and vertical motions are analysed separately.

For a displacement of magnitude r at angle θ above the horizontal, the horizontal component is r cos θ and the vertical component is r sin θ. Here, r is the magnitude of the displacement (m), and θ is the angle of the displacement above the horizontal (° or rad). Components are useful because motion in perpendicular directions can often be analysed independently.

The important difference

Distance is a scalar quantity: the total length of the path travelled by a body. Displacement is the vector change in position from start to finish. Distance follows the route taken. Displacement only looks at the start and end points.

That is why distance can never be negative, while displacement can be positive, negative or zero in one-dimensional problems. Run one full lap of a circular track and your distance is the circumference of the track. Your displacement, however, is zero, because you finish where you started.

Average speed and average velocity begin here

Since distance and displacement are not the same thing, average speed and average velocity are not the same either. Average speed is a scalar quantity that gives total distance travelled per unit time. Average velocity is a vector quantity that gives displacement per unit time.

For the same journey, average speed can be non-zero while average velocity is zero. That happens whenever an object returns to its starting point. In class I usually put it this way: distance remembers the journey; displacement remembers only the final change in position.

Average values

An average value is calculated over a finite time interval. In motion, average speed uses the total distance travelled in that interval, while average velocity uses the displacement during the interval.

You find average acceleration from the overall change in velocity over the chosen interval. It doesn't show every detail of the motion inside that time; it gives the single constant acceleration that would cause the same overall change in velocity in the same time.

Instantaneous values

An instantaneous value is the value at one particular instant, found by looking at an extremely small time interval around that instant. A car speedometer shows instantaneous speed, not the average speed for the whole journey.

On a distance–time graph, the gradient gives speed if the graph shows distance travelled. If it shows displacement along a chosen axis, the gradient gives velocity. For a curved graph, take the gradient of the tangent at the point of interest to find the instantaneous value. Draw a large, clear tangent line; tiny triangles make reading errors worse.

Graphs are a useful way to find physical quantities. The gradient gives the ratio “change in vertical-axis quantity ÷ change in horizontal-axis quantity”. The area under a graph gives the product of the two plotted quantities. So the axes and units are not just decoration — they tell you what the gradient or area means physically.

Velocity–time graphs

On a velocity–time graph, the gradient gives acceleration. A positive gradient means positive acceleration; a negative gradient means negative acceleration. A horizontal line shows constant velocity, so the acceleration is zero.

The signed area under a velocity–time graph gives displacement. Area above the time axis is positive displacement; area below the time axis is negative displacement. If the graph is speed–time instead of velocity–time, the area gives distance travelled.

For a curved velocity–time graph, estimate the area by counting squares. Start by finding the physical value of one square from the axis scales, then count the squares between the curve and the time axis. It is only an estimate, but it can still be a sensible one, and it is often exactly what experimental data calls for.

Measuring motion sensibly

To determine speed experimentally, you need a suitable distance measurement and a suitable time measurement. A slow object can be timed with simple equipment; a fast object may need video analysis, light gates or a sensor. Match the method to the motion. Human reaction time is a large fraction of a 0.2 s event, but a tiny fraction of a 20 s event.

For falling objects, a motion sensor can record a velocity–time graph, whose gradient gives acceleration. Another option is to use an electromagnet and trapdoor timer to measure the fall time for several heights. Plotting height against time squared gives a straighter and more reliable test than trusting one fall only.

When the equations apply

The kinematic equations in the data booklet apply only to uniformly accelerated motion. Uniformly accelerated motion means motion with acceleration constant in both magnitude and direction. Once the acceleration changes, these equations are no longer a safe model for the whole motion.

The four equations are:

s = ((u + v)/2)t, where s is displacement in the chosen direction (m), u is initial velocity in that direction (m s⁻¹), and t is elapsed time (s).

v = u + a**t.

s = u**t + ½a**t².

v² = u² + 2a**s.

Choose a consistent sign convention before you substitute. If upwards is positive, then the downward acceleration due to gravity is negative. If right is positive, motion to the left has negative velocity. Get the signs clear at the start and the algebra will usually take care of the rest.

Where the equations come from

For constant acceleration, the velocity–time graph is a straight line. The gradient gives acceleration; the area gives displacement. That is the whole engine behind these equations.

The equation v = u + a**t comes from the gradient of the straight line. To get s = u**t + ½a**t², add the rectangle under the graph to the triangle above it. The equation s = ((u + v)/2)t uses the mean velocity during constant acceleration. When time is not known, v² = u² + 2a**s is often the useful one.

Choosing an equation

A sensible first step is to list what you know and what you need to find. Then pick the equation that leaves out the unwanted quantity. Don’t begin by searching for a memorised pattern; identify the motion first, and check that the acceleration is constant.

Modelling Newton’s laws

These equations describe what Newton’s laws predict when the resultant force is constant and the mass is constant, since that gives constant acceleration. They work very well for idealised motion: a cart on a uniform slope, a short free fall with negligible air resistance, or a projectile near Earth’s surface when drag can be ignored. They are less reliable for real vehicles, falling leaves, rockets, cyclists and anything where forces or mass change noticeably.

Energy is sometimes the better route here. In projectile problems with no air resistance, conservation of energy can find speed changes due to height changes without working through the detailed time history. Energy alone, though, does not give the full trajectory or time of flight unless it is combined with kinematics. When acceleration is not constant, energy methods often give the cleaner method.

Link to rotational motion

The rotational equations met later are deliberately built to mirror these linear equations. Angular displacement replaces displacement, angular velocity replaces velocity, and angular acceleration replaces acceleration. The mathematics looks familiar because the definitions run in parallel.

Uniform acceleration

Uniform acceleration means acceleration stays constant with time. On a velocity–time graph, it shows up as a straight line. On a displacement–time graph, it gives a curve with a gradient that changes steadily.

Free fall close to Earth is often treated as uniform acceleration, as long as fluid resistance is negligible and the height change is small compared with Earth’s radius. In that model, every object has the same downward acceleration, whatever its mass. It’s a model, not a claim that feathers and stones behave the same way in air.

Non-uniform acceleration

Non-uniform acceleration means acceleration changes with time. The velocity–time graph is curved because its gradient is changing. Motion with fluid resistance is a common example: when speed changes, the drag force changes, and the acceleration changes too.

For non-uniform acceleration, instantaneous acceleration is still found from the gradient of the tangent to a velocity–time graph. Displacement is still the area under the graph. The difference is that the simple constant-acceleration equations cannot be used for the whole motion.

Spreadsheet and numerical models

A spreadsheet can model motion by updating velocity and position in small time steps. With constant acceleration, it produces the same pattern as the kinematic equations. If acceleration depends on speed, as it does with drag, the spreadsheet can still work through the motion step by step. That’s one reason modelling software is useful in modern physics: it lets us investigate motion when the algebra gets awkward.

Measuring gravitational acceleration

One practical way to estimate g is to drop an object and measure velocity as a function of time with a motion sensor. The gradient of the velocity–time graph gives the acceleration. Another method is to release a steel sphere from an electromagnet and time its fall through different heights. Since an object released from rest has s = ½a**t² for constant acceleration, a graph of height against time squared should be a straight line with gradient g/2 if air resistance and timing errors are small.

The projectile model

Projectile motion is two-dimensional motion of a body after launch when the only significant force is gravity. In A.1, quantitative projectile calculations are limited to cases where fluid resistance is absent or negligible.

Near Earth’s surface, we take g as constant and treat Earth’s surface as locally flat. With those assumptions, the horizontal and vertical motions can be handled separately: gravity changes the vertical velocity, but it does not change the horizontal velocity.

Resolving the initial velocity

For a projectile launched at an angle, split the initial velocity into horizontal and vertical components:

uₓ = u cos θ, where uₓ is the initial horizontal component of velocity (m s⁻¹).

uᵧ = u sin θ, where uᵧ is the initial vertical component of velocity (m s⁻¹).

During flight, if fluid resistance is negligible, vₓ = uₓ, where vₓ is the horizontal component of velocity at any later time (m s⁻¹). The vertical component follows the constant acceleration equations; taking upward as positive, vᵧ = uᵧ − g**t, where vᵧ is the vertical component of velocity (m s⁻¹).

The horizontal displacement is sₓ = uₓt, where sₓ is horizontal displacement (m). The vertical displacement is sᵧ = uᵧt − ½g**t², where sᵧ is vertical displacement (m).

Shape of the trajectory

With no fluid resistance, the trajectory is parabolic. For IB, the reasoning is enough: horizontal displacement is proportional to time, while vertical displacement has a term proportional to time squared. You do not need the equation of the trajectory.

At the highest point, the vertical component of velocity is zero. The horizontal component is still there, so the projectile has not “stopped” at the top unless it was launched vertically. Check this carefully in your own reasoning; it’s a common source of mistakes.

Horizontal, angled upward and angled downward launches

For a horizontal launch, uᵧ = 0, and the time of flight comes from the vertical fall. The range is then found by multiplying horizontal velocity by time.

For a launch above the horizontal, the projectile first rises, reaches a highest point where vᵧ = 0, then falls. If it lands at the same height from which it was launched, the time up equals the time down. For a launch below the horizontal, the initial vertical component is already downward, so the flight time is shorter for the same launch height and speed.

For same-level launch and landing, the time to highest point is T = uᵧ/g, where T is time from launch to the highest point (s). The maximum height above the launch point is H = uᵧ²/(2g), where H is maximum vertical height above launch point (m). The range is R = 2T**uₓ, where R is horizontal range (m).

Links to fields and orbits

A mass moving in a uniform gravitational field behaves mathematically like a charged particle entering a uniform electric field at right angles. In both situations, a constant force gives constant acceleration in one direction, while motion in the perpendicular direction continues independently. The result is a parabolic path.

In a non-uniform gravitational field, the motion is different. Over large distances, the gravitational force points towards the centre of the attracting body and changes in strength, so trajectories are not simple parabolas. Circular, elliptical and escape trajectories belong to the broader study of gravitational motion.

Gravitational force makes orbital motion possible because it can keep changing the direction of a satellite’s velocity, providing the inward acceleration needed for a curved path. The satellite is not “force-free”; it is falling around Earth rather than falling straight down.