Question

The acceleration-time graph of a body starting from rest is shown below. Calculate the

(1) Average acceleration

(2) Average velocity

(3) Average speed in the time interval t = 0 to t = 40 s

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Answer 1

(1) during 0 ≤ t ≤ 20 sec

Acceleration , a = 20m/s²

so, v = 20t

initial velocity , u = 20 × 0 = 0 m/s²

during 20s ≤ t ≤ 40s

Acceleration , a = -20 m/s²

velocity at t = 20s , v= 20 × 20 = 400 m/s

so, velocity during 20 ≤ t ≤ 40 , v(t) = 400 - 20t

now, final velocity, v = 400 - 20(40)

= 400 - 800 = -400 m/s

so, average acceleration = change in velocity/time taken

= (-400 - 0)/40 = -10m/s²

(2) average velocity = change in position/time taken

during 0 ≤ t ≤ 20s

velocity function , v = 20t

so, displacement , x = 10t²

and position of body at t = 0s , x = 10 × 0 = 0m

during 20 ≤ t ≤ 40s

velocity function, v = 400 - 20t

so, displacement, x = 400t - 10t²

position of body at t = 40s , x' = 400 × (40) - 10(40)²

= 16000m - 16000m

= 0

so, average velocity = 0/40 = 0 m/s

(3) average speed = total distance/total time taken

total distance , s = 4000 + 400t - 10t²

= 4000 + 400(20)² - 10(20)²

= 8000m

so, average speed = (8000)/40 = 200m/s