A car traveling at a constant speed of 33.4 m/s passes a trooper hidden behind a billboard. One second later the trooper starts the car with a constant acceleration of 3.28 m/s2 How long after the trooper starts the chase does he overtake the speeding car
To find out how long after the trooper starts the chase does he overtake the speeding car, we need to find the time it takes for the trooper's car to catch up with the speeding car.
We can break down the problem into two parts:
1. Finding the position of the speeding car when the trooper starts the chase.
2. Finding the time it takes for the trooper's car to catch up with the speeding car.
First, let's find the position of the speeding car when the trooper starts the chase.
We know that the speeding car has a constant speed of 33.4 m/s. Since the trooper starts the chase one second later, the distance the speeding car covers in that one second is given by:
Distance = Speed * Time
Distance = 33.4 m/s * 1 s
Distance = 33.4 m
So, when the trooper starts the chase, the speeding car is 33.4 meters ahead.
Next, let's find the time it takes for the trooper's car to catch up with the speeding car.
To do this, we will use the equation of motion:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
In this case, the initial velocity of the trooper's car is 0 m/s (since it starts from rest) and the acceleration is 3.28 m/s^2. The distance is the same as the distance covered by the speeding car when the trooper starts the chase, which is 33.4 meters.
33.4 m = 0 m/s * Time + (1/2) * 3.28 m/s^2 * Time^2
Simplifying the equation:
33.4 m = (1/2) * 3.28 m/s^2 * Time^2
Multiplying both sides by 2 to remove the fraction:
66.8 m = 3.28 m/s^2 * Time^2
Dividing both sides by 3.28 m/s^2 to isolate Time^2:
Time^2 = 66.8 m / 3.28 m/s^2
Time^2 = 20.366 s
Taking the square root of both sides to find Time:
Time ≈ √(20.366 s)
Time ≈ 4.51 s
Therefore, it takes approximately 4.51 seconds for the trooper's car to catch up with the speeding car after the trooper starts the chase.