Distance as area under a speed-time graph Problem set A One question at a time

A drone flies at a constant speed of 11 m/s for 14 s. Find the distance it flies.

• Rectangle: width 14 s, height 11 m/s
• \(14\times 11=154\) m

A lorry accelerates steadily from rest to 18 m/s in 9 s. Find the distance it travels in this time.

• Triangle: base 9 s, height 18 m/s
• \(\tfrac12\times 9\times 18=81\) m

A car slows steadily from 24 m/s to 8 m/s in 5 s as it nears a junction. Find the distance it travels in this time.

• Trapezium: parallel sides 24 and 8, width 5
• \(\tfrac12(24+8)\times 5=80\) m

A bus speeds up steadily from rest to 8 m/s in 5 s, keeps 8 m/s until \(t=12\) s, then slows steadily to 4 m/s at \(t=16\) s. Find the total distance travelled in the 16 s.

• Split into shapes at the vertical lines
• Triangle \(\tfrac12(5)(8)=20\), Rectangle \(7\times 8=56\), Trapezium \(\tfrac12(8+4)\times 4=24\)
• Add: \(20 + 56 + 24\) = 100 m

The graph shows the speed of a sled, in m/s, for the first 15 s. Read-off values: \((0,0)\), \((5,2)\), \((10,8)\) and \((15,18)\). Straight chords make triangle A, trapezium B and trapezium C.

1. (a) Find the area of triangle A.

   • \(\tfrac12\times 5\times 2=5\)

2. (b) Find the area of trapezium B.

   • \(\tfrac12(2+8)\times 5=25\)

3. (c) Find the area of trapezium C.

   • \(\tfrac12(8+18)\times 5=65\)

4. (d) Use your answers to estimate the total distance travelled.

   • \(5+25+65=95\) m

5. (e) Is your estimate an over-estimate or an under-estimate? Give a reason.

   • The curve bends upwards, so the chords lie above it.
   • The shapes hold more area than the curve: an over-estimate.

A lift in a tower speeds up steadily from rest to 4 m/s in 3 s, rises at 4 m/s for a further 18 s, then slows steadily to rest in 5 s. Find the height the lift rises.

• Triangle, rectangle, triangle
• \(6\), \(72\), \(10\)
• \(6 + 72 + 10\) = 88 m

A car's speed is measured every 2 seconds. Use four trapeziums to estimate the distance travelled in the 8 seconds.

• Each strip: \(\tfrac12(\text{speed}_1+\text{speed}_2)\times 2\)
• Strips: 6, 16, 22, 25
• Total: 69 m

A train accelerates steadily from rest to 12 m/s in 8 s, travels at 12 m/s for T seconds, then slows steadily to rest in 6 s. The total distance is 300 m. Find T.

• Triangles: \(\tfrac12(8)(12)=48\) and \(\tfrac12(6)(12)=36\)
• Rectangle: \(12T\)
• \(48+12T+36=300\)
• \(12T=216\), so \(T=18\)

Water flows into a tank at a rate of R litres per minute. R rises steadily from 0 to 20 in the first 5 minutes, then stays at 20 for the next 10 minutes.

1. (a) Find the volume of water in the tank after 15 minutes.

   • Triangle \(\tfrac12(5)(20)=50\), rectangle \(10\times 20=200\)
   • Total 250 litres

2. (b) The tank holds 400 litres. The rate stays at 20 litres per minute. How many more minutes until it is full?

   • Still to fill: \(400-250=150\) litres
   • \(150\div 20=7.5\) minutes