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IB Mathematics: Applications and Interpretation SL · 13.5: Integration · 13.5.1

Name Date

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

Show the shapes you split the area into, and the area of each. Give distances in metres unless the question says otherwise.

Difficulty: first skills applying them stretch

Skills

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

    51015246810121416Time (s)Speed (m/s)

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

    24681012246810121416182022Time (s)Speed (m/s)

  3. 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.

    1234567246810121416182022242628Time (s)Speed (m/s)

  4. 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.

    246810121416182468101214Time (s)Speed (m/s)

  5. 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.

    510152468101214161820Time (s)Speed (m/s)
    1. (a) Find the area of triangle A.
    2. (b) Find the area of trapezium B.
    3. (c) Find the area of trapezium C.
    4. (d) Use your answers to estimate the total distance travelled.
    5. (e) Is your estimate an over-estimate or an under-estimate? Give a reason.

In context

  1. 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.

    2468101214161820222426283012345678Time (s)Speed (m/s)

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

    Time (s)02468Speed (m/s)06101213

Stretch

  1. 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.

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

jmaths.xyz · AI SL · 13.5.1 Distance as area under a speed-time graph