Speed and average speed

Speed describes how much distance an object travels per unit of time. A speed of $$5\ \text{m/s}$$ means that the object travels 5 metres each second.

For motion at constant speed, the speed $$v$$, distance travelled $$s$$ and time taken $$t$$ are related by:

If distance is measured in metres and time in seconds, speed is measured in metres per second, written $$\text{m/s}$$.

Average speed when speed varies

An object may travel at different speeds during a journey or stop for part of it. Its average speed describes the total distance travelled per unit of time over the whole journey:

The total time includes any time spent at rest. Average speed does not mean that the object travelled at that speed throughout the journey.

For example, suppose a person travels 120 metres in 20 seconds, rests for 10 seconds, then travels another 80 metres in 20 seconds. The total distance is 200 metres and the total time is 50 seconds, including the rest. The average speed is therefore $$200 \div 50 = 4\ \text{m/s}$$.

Velocity

Velocity is speed in a given direction. It tells us both how fast an object is moving and the direction in which it is moving.

For example, an object travelling at $$5\ \text{m/s}$$ east has a different velocity from one travelling at $$5\ \text{m/s}$$ west. Their speeds are the same, but their directions are different.

An object's velocity therefore changes if its speed changes, its direction changes, or both change.

Distance–time graphs

A distance–time graph shows how the distance travelled by an object changes as time passes. Time is plotted on the horizontal axis and distance travelled on the vertical axis.

To plot a graph from measurements, mark each distance at its corresponding time. A graph can also be sketched from a description of motion: its shape must show how the distance travelled changes during each part of the journey.

What the shape shows

The gradient of a distance–time graph describes how quickly distance travelled increases. A steeper gradient means a greater speed.

• A horizontal section represents an object at rest: time passes, but the distance travelled does not change.
• A straight section with a positive gradient represents constant speed: the object travels equal distances in equal time intervals.
• On a curved section that becomes steeper, the gradient increases, so the object accelerates.
• On a curved section that becomes less steep, the gradient decreases, so the object decelerates.

Speed from a straight-line section

The speed on a straight-line section is its gradient. Choose two points on that section and divide the change in distance by the change in time:

For example, if the distance increases from $$20,\text{m}$$ to $$80,\text{m}$$ over a straight-line section lasting $$12,\text{s}$$, the speed is $$(80-20),\text{m} \div 12,\text{s} = 5,\text{m/s}$$.

The units of the gradient follow from the axis units. Metres divided by seconds give metres per second; kilometres divided by hours give kilometres per hour.

Speed–time graphs

A speed–time graph shows how an object's speed changes as time passes. Time is plotted on the horizontal axis and speed on the vertical axis. Each point on the graph gives the object's speed at a particular time.

To plot a graph from data, choose suitable scales, plot each time and speed pair, then join the points in a way that represents the motion. To sketch a graph from a description, choose the shape of each section to show when the object is at rest, moving at constant speed, speeding up or slowing down.

Reading motion from a speed–time graph

• A section on the time axis shows zero speed: the object is at rest.
• A horizontal section above the time axis shows constant speed.
• A rising section shows increasing speed: the object is accelerating.
• A falling section shows decreasing speed: the object is decelerating.

The steepness of a section shows how quickly the speed changes. A straight rising or falling section has a constant rate of change of speed. A curved section shows that this rate is changing.

Distance from a speed–time graph

The distance travelled during a time interval is the area between the graph and the time axis over that interval. Use only the area between the beginning and end of the interval being considered.

For constant speed, the area is a rectangle. Its height is the speed and its width is the time, so:

For a straight rising or falling section, divide the area under the graph into simple shapes, such as a rectangle and a triangle, and add their areas. A triangle has half the area of a rectangle with the same base and height.

For example, suppose speed increases steadily from $$2\ \text{m/s}$$ to $$6\ \text{m/s}$$ over $$4\ \text{s}$$. The area consists of a rectangle of height $$2\ \text{m/s}$$ and a triangle of height $$4\ \text{m/s}$$. Both have a base of $$4\ \text{s}$$:

Acceleration of free fall

Near the Earth’s surface, an object in free fall accelerates downwards at an approximately constant rate. Free fall assumes that the effect of air resistance is negligible.

The acceleration of free fall, denoted by $$g$$, is approximately $$9.8,\mathrm{m/s^2}$$. This means that, as an object falls downwards, its speed increases by approximately $$9.8,\mathrm{m/s}$$ each second.

For example, an object dropped from rest has a downward speed of approximately $$9.8,\mathrm{m/s}$$ after one second and $$19.6,\mathrm{m/s}$$ after two seconds, provided air resistance is negligible.

Acceleration

Acceleration describes how quickly velocity changes. Velocity includes both speed and direction, so a change in velocity can involve a change in speed, direction, or both.

For a change in velocity $$\Delta v$$ over a time interval $$\Delta t$$, acceleration is:

Here $$a$$ is acceleration and $$\Delta v$$ is the final velocity minus the initial velocity. The unit of acceleration is $$\mathrm{m/s^2}$$: a change in velocity of one metre per second for each second.

With constant acceleration, velocity changes by equal amounts in equal time intervals. With changing acceleration, the rate at which velocity changes varies.

Acceleration on a speed–time graph

For motion in one direction, the gradient of a speed–time graph shows how quickly speed changes. If the direction of motion is taken as positive, the acceleration is:

A straight, sloping section has a constant gradient and therefore represents constant acceleration. A curved section has a changing gradient and represents changing acceleration.

A falling section has a negative gradient: speed decreases, so the object decelerates. When the direction of motion is positive, this deceleration is a negative acceleration, and the negative sign must be used in calculations. For example, if speed falls from $$10\ \mathrm{m/s}$$ to $$4\ \mathrm{m/s}$$ in $$3\ \mathrm{s}$$, the acceleration is $$-2\ \mathrm{m/s^2}$$.

Falling motion and terminal velocity

When an object falls, its weight pulls it downwards. Whether its downward velocity keeps increasing depends on the other forces acting on it.

Falling without resistance

In a uniform gravitational field, an object falling without air or liquid resistance accelerates downwards at the constant gravitational field strength, $$g$$. Its downward velocity increases by equal amounts in equal time intervals.

Falling through air or a liquid

Air or liquid resistance acts upwards on a falling object, opposing its motion. As the object speeds up, resistance generally increases. This reduces the resultant downward force, so the object continues to speed up but its acceleration decreases.

The air or liquid can also exert an upward force called upthrust. When the upward forces, including resistance and any upthrust, balance the object’s weight, the resultant force is zero. The object has then reached terminal velocity.

At terminal velocity, acceleration is zero. The object does not stop: it continues falling at a constant velocity.