Question

A speeding car is traveling along a straight road with a constant speed of 100 km/hour. It passes a police car. The police car starts 5.00 s after the speeder passes. The police car accelerates uniformly (i.e. at constant acceleration) for 10.0 s, at the end of which its speed is 120 km/hour. After that, the police car moves with the constant speed of 120 km/hour till it reaches the speeding car.

a) How far ahead of the police car is the speeding car when the police car stops accelerating?

b) How long after the police car stops accelerating does the it reach the speeding car?

c) How far does the police car travel (from its starting point) before it reaches the speeding car?

d) Sketch the trajectories of the two vehicles in an x vs. t graph. A qualitative plot is sufficient.

Answer 1

a)

Consider the motion of police car

v_f = final speed after accelerating = 120 km/h = 33.33 m/s

v_o = initial speed = 0 m/s

t = time of accelerating = 10 sec

x_p = distance traveled by police car while accelerating

distance traveled by police car while accelerating is given as

x_p = (v_o + v_f )t/2 = (0 + 33.33) (10)/2 = 166.65 m

Consider the motion of speeding car :

v = constant speed of the car = 100 km/h = 27.8 m/s

t' = time of travel of speeding car when police car stops accelerating = 5 + 10 = 15 sec

x_s = distance traveled by speeding car

distance traveled by speeding car is given as

x_s = v t'

x_s = (27.8) (15)

x_s = 417 m

x = distance between police car and speeding car = x_s - x_p = 417 - 166.65 = 250.4 m

So speeding car is ahead of police car by 250.4 m

b)

v_fp = final constant speed of police car = 120 km/h = 33.33 m/s

v_s = constant speed of the speeding car = 100 km/h = 27.8 m/s

t = time taken by police car to catch speeding car after it stops accelerating = ?

x = distance between police car and speeding car = x_s - x_p = 417 - 166.65 = 250.4 m

Using the equation

x = (v_fp - v_s ) t

250.4 = (33.33 - 27.8) t

t = 45.3 sec

c)

distance traveled by police car from starting point is given as

x = x_p + v_fpt

x = 166.65 + (33.33) (45.3)

x = 1676.5 m

d)