Motion: Speed, Velocity and Acceleration
Speed and Velocity
Speed tells you how fast an object is moving. It is a scalar quantity — it has magnitude but no direction. The equation linking distance, speed and time is:
$$s = v \times t$$
Where \(s\) is distance (m), \(v\) is speed (m/s), and \(t\) is time (s). You can rearrange this to find speed (\(v = \frac{s}{t}\)) or time (\(t = \frac{s}{v}\)). Typical speeds to know: walking (~1.5 m/s), running (~3 m/s), cycling (~6 m/s), a car in town (~13 m/s), and the speed of sound in air (~340 m/s).
Velocity is speed in a stated direction. It is a vector quantity. An object travelling in a circle at constant speed has a changing velocity because its direction constantly changes — meaning it is always accelerating, even though its speed stays the same.
Similarly, distance (scalar) measures total ground covered, while displacement (vector) measures how far and in what direction from the start. A runner completing a 400 m lap of a track covers a distance of 400 m but has a displacement of zero — they end up back where they started.
Speed and distance are scalars (magnitude only). Velocity and displacement are vectors (magnitude + direction). An object moving in a circle at constant speed has a changing velocity because direction constantly changes.
Acceleration
Acceleration is the rate at which velocity changes. An object accelerates when it speeds up, slows down, or changes direction. The equation is:
$$a = \frac{v - u}{t}$$
Where \(a\) is acceleration (m/s2), \(v\) is final velocity (m/s), \(u\) is initial velocity (m/s), and \(t\) is time (s). A negative value means the object is decelerating. For example, a car braking from 20 m/s to rest in 5 s has an acceleration of \(a = \frac{0 - 20}{5} = -4 \text{ m/s}^2\).
Question: A cyclist accelerates from 2 m/s to 14 m/s in 6 seconds. Calculate her acceleration.
Step 1: Identify values: \(u = 2 \text{ m/s}\), \(v = 14 \text{ m/s}\), \(t = 6 \text{ s}\).
Step 2: Apply the equation: \(a = \frac{v - u}{t} = \frac{14 - 2}{6} = \frac{12}{6} = 2 \text{ m/s}^2\)
Note: Always include units in your final answer. Forgetting units is one of the most common reasons for losing marks on calculation questions.
Distance-Time Graphs
On a distance-time graph, time is always plotted on the x-axis and distance on the y-axis. The gradient (slope) of the line gives the speed. You can read the shape of the line as follows:
• A horizontal line means the object is stationary (speed = 0).
• A straight diagonal line means constant speed — steeper gradient means greater speed.
• A curve getting steeper means the object is accelerating.
• A curve getting shallower means the object is decelerating.
To find the speed at a specific instant on a curved distance-time graph, draw a tangent to the curve at that point and calculate its gradient.
On a distance-time graph: gradient = speed. On a velocity-time graph: gradient = acceleration. Time is always on the x-axis for motion graphs. Plotting them the wrong way round is a common exam error.
Velocity-Time Graphs
On a velocity-time graph, two key pieces of information can be extracted:
• The gradient equals the acceleration. A steeper slope means greater acceleration. A horizontal line means constant velocity (zero acceleration). A downward slope means deceleration.
• A line below the x-axis means the object is moving in the opposite direction.
The area under the line on a velocity-time graph equals the distance travelled. For a triangle, the area is \(\frac{1}{2} \times \text{base} \times \text{height}\). For a rectangle, it is base × height. For irregular shapes, split into triangles and rectangles and add them together.
To measure the speed of a trolley rolling down a ramp, mark a measured distance on the ramp. Use a stopwatch to time how long the trolley takes to travel between two marked points, then calculate speed using \(v = \frac{s}{t}\). For greater precision, use light gates connected to a data logger — these measure very short time intervals as the trolley interrupts the beam, allowing you to calculate instantaneous speed. Using two light gates at different positions lets you compare speeds and calculate acceleration.
Terminal Velocity
When an object falls through a fluid (such as air or water), two forces act on it: its weight (downwards) and drag (upwards, opposing motion). Initially, weight is much greater than drag, so the object accelerates downwards. As speed increases, drag increases. Eventually, drag equals weight. At this point the resultant force is zero and the object reaches terminal velocity.
A skydiver provides a clear example. After jumping, they accelerate rapidly. As speed builds, air resistance increases until it equals their weight — this is their first terminal velocity (around 55 m/s for a spread-eagled position). When the parachute opens, surface area dramatically increases, causing air resistance to suddenly far exceed weight. The skydiver decelerates until a new, much lower terminal velocity is reached (around 5–7 m/s), safe for landing.
On a velocity-time graph, terminal velocity appears as a curve that starts steep (rapid acceleration) and gradually flattens out to a horizontal line (constant velocity). The curve of a parachute opening would show a sharp decrease in velocity before levelling off again at the lower terminal speed.