An automobile travels along a straight road at 15.65 m/s through a 11.18 m/s speed zone. A police car observed the automobile. At the instant that the two vehicles are abreast of each other, the police car starts to pursue the automobile at a constant acceleration of 1.96 m/s². The motorist noticed the police car in his rear view mirror 12 s after the police car started the pursuit and applied his brakes and decelerates at 3.05 m/s². (Hint: The police will not go against the law.) a) Find the total time required for the police car to overtake the automobile. (12 marks) b) Find the total distance travelled by the police car while overtaking the automobile. (2 marks) c) Find the speed of the police car at the time it overtakes the automobile. (2 marks) d) Find the speed of the automobile at the time it was overtaken by the police car. (2 marks)

A.) Time = 13.75 seconds

B.) Total distance = 339 m

C.) V = 11.18 m/s

D.) V = 10.2 m/s

Explanation: Given that the automobile travels along a straight road at 15.65 m/s through a 11.18 m/s speed zone.

Then,

Initial velocity U of the motorist = 15.65m/s

acceleration a = - 3.05 m/s^2

Initial velocity u of the police man = 11.18 m/s

Acceleration a = 1.96 m/s^2

The police will overtake at distance S as the motorist decelerate and come to rest.

Where V = 0 and a = negative

While the police accelerate.

Using 2nd equation of motion for the motorist and the police

S = ut + 1/2at^2

Since the distance S covered will be the same, so

15.65t - 1/2×3.05t^2 = 11.18t +1/2×1.96t^2

Solve for t by collecting the like terms

15.56t - 1.525t^2 = 11.18t + 0.98t^2

15.56t - 11.18t = 0.98t^2 + 1.525t^2

4.38t = 2.505t^2

t = 4.38/2.505

t = 1.75 seconds approximately

But the motorist noticed the police car in his rear view mirror 12 s after the police car started the pursuit.

Therefore, the total time required for the police car to overtake the automobile will be:

12 + 1.75 = 13.75 seconds

B.) Using the same formula

S = ut + 1/2at^2

Where S = total distance travelled

Substitutes t into the formula

S = 11.18(13.75) + 1/2 × 1.96 (13.75)^2

S = 153.725 + 185.28

S = 339 m approximately

C.) The speed of the police car at the time it overtakes the automobile will be constant = 11.18 m/s

D.) Using first equation of motion

V = U - at

Since the motorist is decelerating

V = 15.65 - 3.05 × 1.75

V = 15.65 - 5.338

V = 10.22 m/s

Therefore, the speed of the automobile at the time it was overtaken by the police car is 10.2 m/ s approximately

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