Final answer:
To catch up to the car moving at 70 km/h, the police motorcycle accelerating at 7.2 m/s² will catch up in approximately 2.7 seconds. At that instant, the motorcycle would be about 26.3 meters behind the car, having traveled about 26.2 meters from its starting point.
Step-by-step explanation:
To solve for the time it takes for the police motorcycle to reach the speed of the car, we first need to convert the car's speed from km/h to m/s. The car is traveling at 70 km/h, which is approximately 19.44 m/s (1 km/h = 0.27778 m/s). The motorcycle is accelerating from rest at 7.2 m/s² until it reaches this speed. The formula to calculate the time when the motorcycle reaches the speed of the car is:
v = u + at
where v is the final velocity, u is the initial velocity (which is 0 since the motorcycle starts from rest), a is the acceleration, and t is the time. Plugging in the values, we get:
19.44 m/s = 0 + (7.2 m/s²) · t
Dividing both sides by 7.2 m/s², we find that t is approximately 2.7 seconds.
To calculate the distance the motorcycle is from the car when it reaches this speed, we need to find the distance travelled by each in this time frame. For the motorcycle, the distance (d) can be calculated using the formula:
d = ut + ½ at²
Substituting the given values, we calculate the distance covered by the motorcycle:
d = 0 · 2.7 s + ½ · 7.2 m/s² · (2.7 s)²
d = ½ · 7.2 m/s² · 7.29 s²
d ≈ 26.2 meters
During the same time, the car travels at a constant speed of 19.44 m/s, so the distance it covers is:
d = v · t
d = 19.44 m/s · 2.7 s
d ≈ 52.5 meters
The motorcycle is therefore approximately 52.5 meters - 26.2 meters = 26.3 meters from the car when it reaches the speed of 70 km/h.