Forces and motion, Year 11: motion graphs and acceleration

speed

提示The gradient is metres gained divided by seconds taken.

为什么Gradient means the rise divided by the run. On these axes the rise is a distance and the run is a time, so the gradient is a distance divided by a time. A steeper line means the object is covering more metres in each second.

It is increasing

提示Compare how much distance is added in each second early on and later.

为什么The gradient of a distance–time graph is the speed, so a line that gets steeper shows a speed that is rising: the car is accelerating. A straight sloping line would mean a steady speed, and a horizontal line would mean the car is not moving.

A straight line through the origin — the car moves at a steady speed

提示Look at how much distance is added in each two-second step.

为什么The car gains the same 3 m in every 2 s, so the points lie on a straight line starting at the origin. An unchanging gradient means an unchanging speed; here it is 3 ÷ 2 = 1.5 m/s. Time goes on the horizontal axis and distance on the vertical axis.

4

提示Find how far the line rises and how far it runs between the two points.

为什么Speed = gradient = change in distance ÷ change in time = (130 − 50) ÷ (30 − 10) = 80 ÷ 20. Always use the changes between two points on the line, not the coordinates of a single point.

8 m/s

提示The tangent is a straight line, so find its gradient in the usual way.

为什么For a curve the gradient is different at every moment, so you draw the straight line that just touches the curve at the time you want and measure the gradient of that line: (32 − 0) ÷ (6 − 2) = 32 ÷ 4. A bigger triangle on the tangent gives a more accurate answer.

2 m/s²

提示First find how much her speed changed, then share that change across the time it took.

为什么Acceleration = change in velocity ÷ time taken, or a = Δv ÷ t; AQA expects this equation to be recalled. Her velocity rose by 6 m/s (from 2 to 8), and 6 ÷ 3 = 2, so she gained 2 m/s in every second. Only the change is divided by the time.

second

提示Read the unit aloud slowly and listen for the word you say twice.

为什么m/s² is short for 'metres per second, per second': a speed (m/s) gained each second. A car accelerating at 3 m/s² from rest does 3 m/s after one second, 6 m/s after two and 9 m/s after three.

Car A

提示Work out how much speed each one gains in a single second.

为什么Car A gains 5 m/s every second (10 ÷ 2); Car B gains only 3 m/s every second (30 ÷ 10). Acceleration compares how quickly speed changes, not how much it changes altogether — B ends up faster simply because it kept going for longer.

Deceleration

提示It is what a driver causes by pressing the brake pedal.

为什么In physics slowing down is still an acceleration — a negative one, because the speed is falling rather than rising. The separate word is handy, but the idea underneath is the same: the velocity is changing, and a force is making it change.

a ~ 3 m/s²

提示Divide the change in velocity by the time, and do not claim more accuracy than the question gives.

为什么The symbol ~ means 'is approximately'. Here a = Δv ÷ t = 27 ÷ 9, and because both figures are rough the answer is given as about 3 m/s². Estimating everyday accelerations like this is a GCSE skill: a few m/s² is typical for a car pulling away briskly.

acceleration

提示The gradient here is metres per second gained, divided by seconds taken.

为什么On these axes the rise is a change in velocity and the run is a time, so the gradient is change in velocity ÷ time. A steeper line means the velocity is changing more quickly, and a line sloping downwards means the object is slowing.

2

提示Find the change in velocity between the two points and the time it took.

为什么Acceleration = gradient = change in velocity ÷ time taken = (12 − 2) ÷ (5 − 0) = 10 ÷ 5. The line does not start at zero velocity, so it is the change of 10 m/s that matters, not the final 12 m/s.

It moves at a steady 8 m/s, with zero acceleration

提示Read the vertical axis label again before you answer.

为什么A horizontal line on these axes means the velocity is not changing: the gradient, and so the acceleration, is zero. The object is still moving, covering 8 m in every second. It would only be at rest if the line lay along the time axis, at 0 m/s.

−3 m/s² (a deceleration of 3 m/s²)

提示Work out final minus initial velocity, and keep the sign.

为什么Acceleration = change in velocity ÷ time = (6 − 18) ÷ 4 = −12 ÷ 4. The minus sign shows the velocity is falling: a line that slopes downwards has a negative gradient, and the object is decelerating.

The distance travelled (or the displacement)

提示Multiply the units of the two axes together and see what you get.

为什么Height × width on this graph is a velocity × a time, which is a distance: metres per second × seconds gives metres. So the area under the line, found by splitting it into rectangles and triangles or by counting squares, is how far the object has gone.

80 m

提示Split the shape under the line into a triangle and a rectangle.

为什么Distance is the area under the line. The triangle for the first 4 s has area ½ × 4 × 10 = 20 m. The rectangle from 4 s to 10 s has area 6 × 10 = 60 m. Adding the two gives the total distance.