Speed, Velocity and Acceleration
Speed and Distance
Speed is a scalar quantity — it tells you how fast something is moving but not the direction. The equation for speed is:
$$s = v t$$
Where \(s\) is distance in metres, \(v\) is speed in m/s, and \(t\) is time in seconds. This can be rearranged to \(v = \frac{s}{t}\).
Typical speeds: walking ~1.5 m/s, running ~3 m/s, cycling ~6 m/s, car in town ~13 m/s, car on motorway ~30 m/s, train ~55 m/s, aeroplane ~250 m/s, sound in air ~330 m/s.
Velocity
Velocity is a vector quantity. It has both magnitude (speed) and direction. An object moving in a circle at constant speed has a changing velocity because its direction is always changing.
The difference between speed and velocity matters in physics: two cars travelling at 30 m/s in opposite directions have the same speed but different velocities.
Acceleration
Acceleration is how quickly an object’s velocity changes. The equation is:
$$a = \frac{\Delta v}{t} = \frac{v - u}{t}$$
Where \(a\) is acceleration in m/s\(^2\), \(v\) is final velocity, \(u\) is initial velocity, and \(t\) is time in seconds. A negative acceleration means the object is decelerating (slowing down).
For uniform (constant) acceleration, use the equation: \(v^2 = u^2 + 2as\), where \(s\) is the distance travelled. This is particularly useful when time is not given in the question.
A car accelerates from 10 m/s to 30 m/s in 5 seconds. Calculate the acceleration.
\(a = \frac{v - u}{t} = \frac{30 - 10}{5} = \frac{20}{5} = 4\) m/s\(^2\)
Distance-Time Graphs
A distance-time graph shows how far an object has travelled over time. The gradient (slope) of the line gives the speed.
A horizontal line means the object is stationary. A straight diagonal line means constant speed. A curve getting steeper means the object is accelerating. The steeper the line, the faster the object is moving.
To find the speed at a specific moment on a curved distance-time graph, draw a tangent to the curve at that point. Then calculate the gradient of the tangent: \(\text{speed} = \frac{\text{change in distance}}{\text{change in time}}\).
Velocity-Time Graphs
A velocity-time graph shows how velocity changes over time. The gradient gives the acceleration. At Higher tier, the area under the line also gives the total distance travelled.
A horizontal line means constant velocity (no acceleration). A line sloping upwards means acceleration. A line sloping downwards means deceleration. The area under the graph can be found by splitting it into rectangles and triangles.
Use a dynamics trolley on a ramp with a light gate or ticker tape timer. Vary the force (by changing the mass on a pulley) while keeping the total mass constant. Measure the acceleration for each force. Plot a graph of force against acceleration — a straight line through the origin shows that \(F = ma\).