Motion: Mathematical Problems

Math prob (a): The driver of a car travelling at 72 kmph observes the light 300 m ahead of him turning red. The traffic light is timed to remain red for 20 s before it turns green. If the motorist wishes to passes the light without stopping to wait for it to turn green. Determine

(i) the required uniform acceleration of the car.

(ii) the speed with which the motorist crosses the traffic light.

Ans: Data: u = 72 kmph = 20 ms^-1; S = 300 m; t = 20 s; a = ? v = ?

(i) S = ut + ½ at²

Or, 300 = (20*20) + ½ a (20)²

Then, a = – 0.5 ms^-2

(ii) v = u + at = 20 – (0.5 *20) = 10 ms^-1

Math prob (b): A stone is dropped from the top of the tower 50 m high. At the same time another stone is thrown up from the foot of the tower with a velocity of 25 ms^-1. At what distance from the top and after how much time the stones cross each other?

Data: Height of the tower = 50 m; u₁ = 0 ; u₂ = 25 ms^-1

Let s₁ and s₂ be the distances travelled by the two stones at the time of crossing (t). Therefore S₁ + S₂ = 50m; s = ?: t = ?

Ans: For I stone: S₁ = ½ gt²

For II stone: S₂ = u₂t + – ½ gt²

Therefore, S₁ + S₂ = 50 = ½ gt² + 25 t – ½ gt²

t = 2 seconds

so, S₁ = ½ gt² = ½ (9.8) 2² = 19.6 m