Acceleration, Velocity-Time Graphs and Terminal Velocity (Part 1/3) Revision Notes | GCSE AQA Higher Physics

1. Why Velocity-Time Graphs Matter

Picture a car accelerating from traffic lights, then cruising at constant speed, then braking to a stop. How do we capture this changing motion mathematically? Distance-time graphs show position, but they hide the real story of how speed changes moment by moment.

Velocity-time graphs reveal the hidden patterns. They show not just how fast something moves, but how that speed changes - and that rate of change has a special name: acceleration. Every time you feel pushed back in your seat during takeoff, or thrown forward when braking, you're experiencing what these graphs measure.

This connects directly to forces. When forces are unbalanced, objects accelerate. When they balance out, motion becomes constant. The graphs become our window into the invisible force battles happening around moving objects.

2. The Language of Motion

The key insight is that the steeper the line, the greater the acceleration. A horizontal line means zero acceleration - constant velocity. A downward slope means negative acceleration - the object is slowing down.

Think of a skydiver. Initially they accelerate downwards at 9.8 m/s² due to gravity. But as speed increases, air resistance grows stronger. Eventually these forces balance, acceleration drops to zero, and velocity becomes constant - that's terminal velocity.

3. The Acceleration Equation

The triangle symbol Δ (delta) means 'change in'. So Δv = final velocity - initial velocity. If a car goes from 5 m/s to 20 m/s, then Δv = 20 - 5 = 15 m/s. The time t is how long this change took.

Units are crucial here. Acceleration has units of m/s² because we're dividing velocity (m/s) by time (s). This gives us m/s/s, which we write as m/s². Think of it as 'metres per second, per second' - how much the velocity changes each second.

Worked example: Calculating acceleration from velocity change

A cyclist accelerates from rest to 12 m/s in 8 seconds. Calculate the acceleration. [3 marks]

1. $a = \frac{\Delta v}{t}$ — [1 mark for correct formula]
2. $\Delta v = 12 - 0 = 12 \text{ m/s}$ — From rest means initial velocity = 0 m/s
3. $a = \frac{12}{8} = 1.5 \text{ m/s}^2$ — [1 mark for substitution, 1 mark for answer with units]

Answer: 1.5 m/s²

4. Reading Velocity-Time Graphs

Velocity-time graphs tell stories about motion. The y-axis shows velocity (usually in m/s), the x-axis shows time (in s). Every point on the line tells you the velocity at that exact moment. But the real power lies in reading the shape of the line.

A horizontal line means constant velocity - zero acceleration. The object moves at steady speed. A sloping line upwards means positive acceleration - the object speeds up. A sloping line downwards means negative acceleration - the object slows down. The steeper the slope, the greater the acceleration.

Here's the key insight: the gradient (slope) of a velocity-time graph equals the acceleration. To find gradient, you calculate rise over run: change in y-axis divided by change in x-axis. On a v-t graph, this gives you change in velocity divided by change in time - which is exactly the acceleration equation.

Real motion often shows curved lines. A car pulling away from traffic lights doesn't instantly reach maximum acceleration - it builds up gradually. This creates a curved line that starts steep (high acceleration) then levels off (lower acceleration) as the car approaches its cruising speed.

5. Terminal Velocity and Force Balance

When objects fall through air, they don't accelerate forever. Initially, gravity dominates and the object accelerates downwards at 9.8 m/s². But as velocity increases, air resistance grows stronger. Eventually these forces balance out, acceleration drops to zero, and velocity becomes constant.

This creates a distinctive velocity-time graph shape. It starts with a steep upward slope (high acceleration due to gravity), then gradually curves over as air resistance builds up, finally becoming horizontal when terminal velocity is reached. The object continues falling, but at constant speed.

Different objects reach different terminal velocities. A feather reaches terminal velocity quickly at low speed because its large surface area creates lots of air resistance. A steel ball reaches terminal velocity much later at high speed because its small surface area creates less air resistance relative to its weight.

The key physics principle is force balance. When upward forces (air resistance, drag) equal downward forces (weight, gravity), the net force becomes zero. Newton's First Law tells us that zero net force means zero acceleration. Hence constant velocity - terminal velocity.

6. Exam Technique and Common Mistakes

Velocity-time graph questions are mark-winners if you know the technique. Always start by identifying what the axes represent - velocity on y-axis, time on x-axis. Then look at the shape: horizontal means constant velocity, sloping means acceleration, curved means changing acceleration.

For gradient calculations, examiners want to see your working. Draw a large triangle on the graph, clearly mark the rise and run values with units, then calculate gradient = rise/run. Always include units in your final answer - acceleration must be in m/s².

When describing motion from graphs, use precise language. Don't say 'the object goes faster' - say 'the object accelerates' or 'velocity increases'. Don't say 'it stops accelerating' - say 'acceleration becomes zero' or 'velocity becomes constant'. Examiners reward scientific precision.