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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 B

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

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

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

    51015202512345678Time (s)Speed (m/s)

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

    246810121424681012141618Time (s)Speed (m/s)

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

    1234567892468101214Time (s)Speed (m/s)

  4. 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.

    2468101214162468101214Time (s)Speed (m/s)

  5. 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.

    48121620242832362468101214Time (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 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.

    1020304050607012345678Time (s)Speed (m/s)

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

    Time (s)036912Speed (m/s)047910

Stretch

  1. 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.

  2. 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.
    2. (b) The bucket holds 150 litres. The rate stays at 12 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