What kinematics is trying to do

Kinematics is the part of mechanics that describes motion without explaining the forces that cause it. That last phrase is doing real work. In this topic, we ask where is it?, how fast is it moving?, and how is that motion changing? The why of the motion changing is left for forces in A.2.

We can describe a moving body quantitatively using position, velocity and acceleration. Position is a vector quantity that specifies 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 these, a sentence like “the ball is $3\ \text{m}$ away” isn't really physics yet.

For one-dimensional motion, a single axis is enough to give position. In two dimensions, you need two coordinates; in three dimensions, three coordinates are needed. You may choose the axes, but after that, stay consistent. Projectile problems, for instance, are usually much cleaner when horizontal and vertical axes are used.

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 alone 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 one-dimensional vectors, though, a sign is often used as a direction code: if right is chosen as positive, then a velocity of $-2\ \text{m s}^{-1}$ means $2\ \text{m s}^{-1}$ 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 show the size of the effect, while the words make clear what motion is being described.

The central modelling idea in this topic is simple: position changes with time. If we know the pattern of that change, we can predict the future position. That prediction depends on the model assumptions. 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,

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

Speed is a scalar quantity that tells you how quickly distance is travelled. For a steady journey, $speed = \text{distance travelled} \div \text{time taken}$. 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 that tells you how quickly velocity changes with time:

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 does not mean “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 necessarily in the direction of motion. An object moving to the right while slowing down has acceleration to the left. An object moving to the left while slowing down has acceleration to the right. This sign logic is small, but it prevents a lot of mistakes.

Near Earth’s surface, freely falling bodies are often modelled using gravitational acceleration, which is an acceleration caused by Earth’s gravitational field. We usually use

or $9.8\,m\,s^{-2}$

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 get to school; another might somehow go straight from home to school. If they start and finish at the same places, their displacement is the same.

In one dimension, displacement is positive or negative depending on the direction convention. If upwards is positive, a ball that starts at $1.2\text{ m}$ above the ground and later reaches $0.9\text{ m}$ above the ground has a displacement of $-0.3\text{ m}$. The minus sign shows that the change in position is downward.

Components of displacement

A vector can be resolved into perpendicular components. For movement on a slope, the displacement along the slope can be split into a horizontal component and a vertical component. This isn’t just mathematical decoration: it’s the same habit needed later for projectile motion, where horizontal and vertical motions are analysed separately.

For a displacement of magnitude $r$ at angle $\theta$ above the horizontal, the horizontal component is $r\cos\theta$ and the vertical component is $r\sin\theta$, where $r$ is the magnitude of the displacement (m) and $\theta$ is the angle of the displacement above the horizontal ($^\circ$ 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 a body travels. Displacement is the vector change in position from start to finish. Distance follows the route taken. Displacement cares only about 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, though, is zero, because you finish where you started.

Average speed and average velocity begin here

Since distance and displacement are different, average speed and average velocity differ too. Average speed is a scalar quantity: total distance travelled per unit time. Average velocity is a vector quantity: 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 like this: 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 during that interval, while average velocity uses displacement during that interval.

Average acceleration comes from the overall change in velocity across the chosen interval. It doesn't describe every little change that happened inside the interval. Instead, it gives the single constant acceleration that would produce the same overall change in velocity in the same time.

Instantaneous values

An instantaneous value is a 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 the graph shows displacement along a chosen axis, the gradient gives velocity. For a curved graph, find the instantaneous value from the gradient of the tangent at the point of interest. Draw a large, clear tangent line; tiny triangles make reading errors much worse.

Graphs are often the neatest way to determine physical quantities. The gradient of a graph 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. That is why graph axes and units are not decoration — they tell you what the gradient or area physically means.

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 rather than velocity–time, the area gives distance travelled.

For a curved velocity–time graph, you can estimate the area by counting squares. First work out the physical value of one square from the axis scales, then count the squares between the curve and the time axis. It is an estimate, but a sensible one, and often exactly what experimental data requires.

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. The method has to fit the motion. A human reaction time is a large fraction of a $0.2\,\text{s}$ event, but a tiny fraction of a $20\,\text{s}$ 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 is motion in which acceleration is constant in magnitude and direction. Once acceleration changes, these equations are no longer a safe model for the whole motion.

The four equations are:

$v = u + at.$

$s = ut + \frac{1}{2}at^2.$

$v^2 = u^2 + 2as.$

Set a consistent sign convention before you substitute. If upwards is positive, then downward acceleration due to gravity is negative. If right is positive, motion to the left has negative velocity. The algebra will usually behave if the signs are chosen clearly at the start.

Where the equations come from

For constant acceleration, the velocity–time graph is a straight line. Its gradient gives acceleration and its area gives displacement. That’s the whole engine behind these equations.

The equation $v = u + at$ comes from the straight-line gradient. To get $s = ut + \frac{1}{2}at^2$, add the rectangle under the graph to the triangle above it. The equation $s = \frac{u + v}{2}t$ uses the mean velocity during constant acceleration. The equation $v^2 = u^2 + 2as$ is handy when time is not known.

Choosing an equation

A neat way to proceed is to list the quantities you know and the one you want. Then choose the equation that leaves out the unwanted quantity. Don’t start by hunting for a memorised pattern; start by identifying the physics of the motion and checking that acceleration is constant.

Modelling Newton’s laws

These equations model the consequences of Newton’s laws when the resultant force is constant and the mass is constant, because 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 weaker for real vehicles, falling leaves, rockets, cyclists and anything where forces or mass change noticeably.

Sometimes energy gives the cleaner route. For projectile problems where air resistance is absent, conservation of energy can find speed changes due to height changes without calculating the detailed time history. But energy alone does not give the full trajectory or time of flight unless combined with kinematics. When acceleration is not constant, energy methods often become the cleaner method.

Link to rotational motion

The rotational equations met later are deliberately built to mirror these linear equations. Displacement is replaced by angular displacement, velocity by angular velocity, and acceleration by angular acceleration. The mathematics looks familiar because the definitions are 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, provided 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, though; it doesn’t say that feathers and stones behave the same in air.

Non-uniform acceleration

Non-uniform acceleration means acceleration changes with time. The velocity–time graph is then a curve, since its gradient is changing. Motion with fluid resistance is a common example: when speed changes, the drag force changes, so the acceleration changes as well.

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 over 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 method for estimating g is to drop an object and measure velocity against 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 $s = \frac{1}{2}at^2$ for constant acceleration, a graph of height against time squared should be a straight line with gradient $g/2$, as long as 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 the surface as locally flat. With those assumptions, the horizontal and vertical parts of the motion 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:

During flight, if fluid resistance is negligible,

The vertical component follows the constant acceleration equations. Taking upward as positive,

Horizontal displacement is

Vertical displacement is

Shape of the trajectory

Without fluid resistance, the trajectory is parabolic. The IB reason is straightforward: 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. It’s a common place for mistakes in reasoning.

Horizontal, angled upward and angled downward launches

For a horizontal launch, $u_y = 0$, so the time of flight comes from the vertical fall. Once you have that time, the range comes from horizontal velocity multiplied by time.

For a launch above the horizontal, the projectile first rises, reaches a highest point where $v_y = 0$, 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

The maximum height above the launch point is

The range is

Links to fields and orbits

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

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.

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

What fluid resistance means

Fluid resistance is the resistive effect produced by a gas or liquid on a body moving through it. Air resistance is the usual projectile example; in water, the same idea appears as water resistance.

A fluid is any substance that can flow and does not have a fixed shape of its own, so both gases and liquids are fluids. As an object moves, it transfers energy to the surrounding fluid and feels a resistive force in the opposite direction to its motion. Usually, the faster it moves, the larger this resistive effect becomes.

Terminal speed

Terminal speed is the constant speed a falling body reaches when the upward resistive force from the fluid balances the downward gravitational force. At terminal speed, the resultant force is zero and the acceleration is zero, even though the object is still moving.

For a body falling vertically from rest, the acceleration starts close to $g$ if the initial drag is small. As the speed rises, drag increases, so the acceleration falls. The speed–time graph gradually levels off, and once terminal speed is reached, the distance–time graph becomes closer to a straight line.

Projectiles with fluid resistance

Once fluid resistance is included, projectile motion is no longer the neat parabolic model. The resistive force acts opposite to the velocity, which gives it both horizontal and vertical effects. The projectile has horizontal deceleration, its vertical acceleration is no longer constant, and its speed is reduced throughout the motion compared with the no-drag case.

Learn the qualitative effects as one linked story:

• the range is reduced;
• the maximum height is reduced;
• the trajectory is no longer parabolic and is usually steeper on the way down;
• the time of flight changes and depends on the launch conditions and drag strength;
• the velocity at each point is smaller than in the no-drag model for the same launch conditions;
• the acceleration is not constant because the drag force changes as the speed and direction change;
• terminal speed may become relevant on the downward part of the motion.

A quantitative treatment of projectile motion with fluid resistance is beyond A.1. In this topic, if a projectile question needs numbers, fluid resistance will be absent or negligible. If fluid resistance is included, expect a qualitative explanation, graph interpretation or modelling discussion rather than a simple suvat calculation.