a. How much time elapses before the motorcycle is moving as fast as the car?
b. How far is the motorcycle from the car when it reaches this speed?
(a) The time elapses before the motorcycle is moving as fast as the car is 2.6296 seconds.
(b) The motorcycle is 25.93 meters far from the car when it reaches this speed.
Understanding Acceleration
Acceleration is defined as the rate of change of velocity with time. It is measured in meters per second squared (m/s²). In this problem, we are given that a car is traveling at a steady speed of 71 km/h in a 50 km/h zone, while a police motorcycle takes off at the instant the car passes it, accelerating at a constant rate of 7.5 m/s².
To solve for the time it takes for the motorcycle to reach the speed of the car, we need to convert the speed of the car to meters per second. Given that the speed of the car is 71 km/h, we can convert this to meters per second by multiplying by 5/18:
Speed of the car = 71 km/h = 71 × 5/18 m/s = 19.722 m/s
Next, we can calculate the time taken by the motorcycle to be as fast as the car by dividing the speed of the car by the acceleration of the police motorcycle:
Time taken = Speed of the car / Acceleration = 19.722 m/s / 7.5 m/s² = 2.6296 seconds
Now, to find the distance between the motorcycle and the car when the motorcycle reaches the speed of the car, we use the formula for distance traveled during accelerated motion:
Distance = (Initial velocity × Time) - (1/2 × Acceleration × Time²)
Distance = (19.722 × 2.6296) - (1/2 × 7.5 × 2.6296²) = 25.93 meters
In conclusion, it takes 2.6296 seconds for the motorcycle to move as fast as the car, and at that moment, the motorcycle is 25.93 meters away from the car.