Distance as area under a speed-time graph Problem set B Worked solutions

A conveyor belt carries a parcel at a constant speed of 3 m/s for 25 s. Find the distance the parcel travels.

• Rectangle: width 25 s, height 3 m/s
• \(25\times 3=75\) m

A skier accelerates steadily from rest to 14 m/s in 12 s. Find the distance she travels in this time.

• Triangle: base 12 s, height 14 m/s
• \(\tfrac12\times 12\times 14=84\) m

A cyclist speeds up steadily from 4 m/s to 10 m/s in 7 s. Find the distance she travels in this time.

• Trapezium: parallel sides 4 and 10, width 7
• \(\tfrac12(4+10)\times 7=49\) m

A trolley moves at 6 m/s for 4 s, speeds up steadily to 10 m/s at \(t=10\) s, then keeps 10 m/s until \(t=14\) s. Find the total distance travelled in the 14 s.

• Split into shapes at the vertical lines
• Rectangle \(4\times 6=24\), Trapezium \(\tfrac12(6+10)\times 6=48\), Rectangle \(4\times 10=40\)
• Add: \(24 + 48 + 40\) = 112 m

The graph shows the speed of a train leaving a station, in m/s, for the first 36 s. Read-off values: \((0,0)\), \((9,6)\), \((25,10)\) and \((36,12)\). Straight chords make triangle A, trapezium B and trapezium C.

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

   • \(\tfrac12\times 9\times 6=27\)

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

   • \(\tfrac12(6+10)\times 16=128\)

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

   • \(\tfrac12(10+12)\times 11=121\)

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

   • \(27+128+121=276\) m

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

   • The curve bends downwards, so the chords lie below it.
   • The shapes hold less area than the curve: an under-estimate.

A ferry leaves a harbour: its speed rises steadily from 0 to 6 m/s in 10 s, stays at 6 m/s for 40 s, then falls steadily to 0 in 20 s as it docks. Find the distance the ferry sails.

• Triangle, rectangle, triangle
• \(30\), \(240\), \(60\)
• \(30 + 240 + 60\) = 330 m

A runner's speed is measured every 3 seconds. Use four trapeziums to estimate the distance run in the 12 seconds.

• Each strip: \(\tfrac12(\text{speed}_1+\text{speed}_2)\times 3\)
• Strips: 6, 16.5, 24, 28.5
• Total: 75 m

A lift car accelerates steadily from rest to 15 m/s in 6 s, travels at 15 m/s for T seconds, then slows steadily to rest in 10 s. The total distance is 420 m. Find T.

• Triangles: \(\tfrac12(6)(15)=45\) and \(\tfrac12(10)(15)=75\)
• Rectangle: \(15T\)
• \(45+15T+75=420\)
• \(15T=300\), so \(T=20\)

Rain runs into a bucket at a rate of R litres per minute. R rises steadily from 0 to 12 in the first 4 minutes, then stays at 12 for the next 6 minutes.

1. (a) Find the volume of water in the bucket after 10 minutes.

   • Triangle \(\tfrac12(4)(12)=24\), rectangle \(6\times 12=72\)
   • Total 96 litres

2. (b) The bucket holds 150 litres. The rate stays at 12 litres per minute. How many more minutes until it is full?

   • Still to fill: \(150-96=54\) litres
   • \(54\div 12=4.5\) minutes