The area under the line — answer key

1. MQ10 · Practice

Options as numbered on the pupil sheet: 1 — Speeding up; 2 — Constant speed; 3 — Slowing down; 4 — At rest.

Model answer. A: speeding up, because the speed rises from 0 to 8 m/s. B: constant speed, because the line is flat at 8 m/s. C: slowing down, because the speed falls to zero. D: at rest, because the speed is zero.

A full answer includes

• I matched A: speeding up; B: constant speed; C: slowing down; D: at rest.

Also accept

• A speeding up; B constant speed; C slowing down; D at rest.

Watch for

• “B is at rest because the line is flat.” — On a speed–time graph a flat line above the axis means the speed stays at 8 m/s: the scooter keeps moving. At rest is a line along the axis, like D.
• “C means the scooter is coming back.” — A speed–time graph doesn't show direction. A falling line means the speed is getting smaller: slowing down.

2. MQ11 · Practice

Model answer. The area under the flat line is a rectangle 0.30 m/s high and 4.0 s wide: 0.30 m/s × 4.0 s = 1.2 m.

Checked automatically

• Distance travelled: 1.2 m (within ±2%)
  • Common wrong answer 13 : You divided. The area of the rectangle is speed × time.
  • Common wrong answer 0.60 : That is half the rectangle. Use ½ only for triangles.

A full answer includes

• I found the area of the rectangle: 0.30 m/s × 4.0 s = 1.2 m.

Also accept

• 1.2 m

Watch for

• “Distance = 4.0 ÷ 0.30 = 13 m.” — The area of a rectangle is height × width, not width ÷ height. Distance = speed × time.
• “Use ½ × 4.0 × 0.30 = 0.60 m.” — The ½ is for a triangle. Under a flat line the shape is a rectangle.

3. MQ12 · Practice

Model answer. The area under the rising line is a triangle with base 4.0 s and height 6.0 m/s: ½ × 4.0 s × 6.0 m/s = 12 m. (Her average speed is half her top speed, 3.0 m/s, and 3.0 m/s × 4.0 s = 12 m.)

Checked automatically

• Distance travelled in 4.0 s: 12 m (within ±2%)
  • Common wrong answer 24 : You found the whole rectangle. Under a line rising from zero the shape is a triangle: take ½.
  • Common wrong answer 1.5 : You divided the speed by the time. The distance is the area under the line.

A full answer includes

• I used the area of a triangle: ½ × 4.0 s × 6.0 m/s = 12 m.

Also accept

• 12 m

Watch for

• “6.0 m/s × 4.0 s = 24 m.” — That would be true only if she rode at 6.0 m/s the whole time. She started from rest, so the shape is a triangle: take half.
• “6.0 ÷ 4.0 = 1.5 m.” — Dividing gives how quickly her speed rises (1.5 m/s each second), not the distance. Use the area.

4. MQ17 · Practice

Model answer. (a) Time / s across (0 to 6 s), speed / m/s up (0 to 0.30 m/s); seven crosses: a straight rising line from (0, 0) to (3 s, 0.30 m/s), then a flat line at 0.30 m/s to 6 s. (b) 0–3 s: the speed rises by 0.10 m/s every second, so the trolley speeds up evenly from rest. 3–6 s: the speed stays at 0.30 m/s, a constant speed. (c) The area under the first 3 s is a triangle: ½ × 3 s × 0.30 m/s = 0.45 m.

A full answer includes

• (a) My axes are labelled time / s and speed / m/s, the points are small crosses, and I joined them with straight lines part by part.
• (b) 0–3 s: speeding up evenly from rest to 0.30 m/s; 3–6 s: constant speed of 0.30 m/s.
• (c) Triangle: ½ × 3 s × 0.30 m/s = 0.45 m.

Also accept

• (b) speeding up evenly, then constant speed; (c) 0.45 m

Watch for

• “(c) 0.30 m/s × 3 s = 0.90 m.” — The trolley was only at 0.30 m/s at the end of the 3 s. The shape under the line is a triangle: use ½ × base × height.
• “(b) 3–6 s: it has stopped, because the line is flat.” — On a speed–time graph a flat line above the axis means a constant speed. Only a line along the time axis means at rest.

5. MQ13 · Practice

Model answer. c. On a distance–time graph a horizontal line means the distance doesn't change, so the object is at rest. On a speed–time graph it means the speed doesn't change, so the object keeps moving at a constant speed.

A full answer includes

• I chose c: distance–time, at rest; speed–time, constant speed.
• I explained that a constant distance means not moving, while a constant speed means moving at the same speed.

Also accept

• c; constant distance = at rest, constant speed = still moving.

Watch for

• “Both mean at rest.” — On a speed–time graph the line is above zero speed: the object is moving, just not changing speed.
• “Both mean constant speed.” — On a distance–time graph a flat line means the distance stays the same: the object isn't moving.

6. MQ14 · Practice

Model answer. Speeding up: ½ × 10 s × 10 m/s = 50 m. Constant speed: 30 s × 10 m/s = 300 m. Slowing down: ½ × 20 s × 10 m/s = 100 m. Total distance = 50 + 300 + 100 = 450 m. (As one trapezium: ½ × (30 s + 60 s) × 10 m/s = 450 m.)

Checked automatically

• Distance between the stops: 450 m (within ±2%)
  • Common wrong answer 600 : That treats the whole 60 s as 10 m/s, or forgets the ½ for the triangles. Split the area into shapes.
  • Common wrong answer 300 : That is only the rectangle. Add the triangles for speeding up and slowing down.

A full answer includes

• I found the three areas: ½ × 10 s × 10 m/s = 50 m; 30 s × 10 m/s = 300 m; ½ × 20 s × 10 m/s = 100 m.
• I added them: 450 m.

Also accept

• 450 m

Watch for

• “10 m/s × 60 s = 600 m.” — The bus was only at 10 m/s for 30 s. While it sped up and slowed down its speed was lower: those parts are triangles.
• “Only the flat part counts: 300 m.” — The bus also moved while it sped up and slowed down. Add the two triangles.

7. MQ15 · Practice

Model answer. (a) Time / s across, speed / m/s up: a straight line from (0, 0) to (2.0 s, 2.0 m/s), flat to (8.0 s, 2.0 m/s), then straight down to (10.0 s, 0). (b) Distance = area = ½ × 2.0 s × 2.0 m/s + 6.0 s × 2.0 m/s + ½ × 2.0 s × 2.0 m/s = 2.0 m + 12 m + 2.0 m = 16 m.

A full answer includes

• (a) Labelled axes; a straight rise from 0 to 2.0 m/s at 2.0 s, flat at 2.0 m/s to 8.0 s, a straight fall to 0 at 10.0 s.
• (b) ½ × 2.0 × 2.0 + 6.0 × 2.0 + ½ × 2.0 × 2.0 = 2.0 + 12 + 2.0 = 16 m.

Also accept

• (b) 16 m

Watch for

• “2.0 m/s × 10 s = 20 m.” — The lift was below 2.0 m/s while it sped up and slowed down. Use triangles for those parts.
• “Draw the flat part along the time axis.” — Along the axis means zero speed. The lift moves at 2.0 m/s, so the flat part is at 2.0 m/s.

8. MT2 · Apply

Model answer. (a) Speeding up: ½ × 40 s × 20 m/s = 400 m. Steady: 120 s × 20 m/s = 2400 m. Slowing down: ½ × 30 s × 20 m/s = 300 m. Distance = 3100 m (3.1 km). (b) Average speed = 3100 m ÷ 190 s = 16 m/s. (c) While it speeds up, the distance–time line curves upwards, getting steeper; at the steady speed it is a straight rising line; while it slows down it curves over, getting flatter until it is flat when the train stops.

A full answer includes

• (a) ½ × 40 × 20 + 120 × 20 + ½ × 30 × 20 = 400 + 2400 + 300 = 3100 m.
• (b) 3100 m ÷ 190 s = 16 m/s (16.3 m/s).
• (c) A curve getting steeper, then a straight rising line, then a curve getting flatter (ending flat).

Also accept

• (a) 3100 m; (b) 16 m/s; (c) steeper curve, straight line, flatter curve

Watch for

• “(b) Average speed = (0 + 20) ÷ 2 = 10 m/s.” — The train spent most of the time at 20 m/s. Average speed = total distance ÷ total time.
• “(c) The distance–time graph rises, is flat, then falls, like the speed–time graph.” — The train keeps moving forwards, so its distance never falls. Where the speed is constant the distance–time line is straight, not flat.