A police car is traveling at a velocity of 15.0 m/s due north, when a car zooms by at a constant velocity of 40.0 m/s due north. After a reaction time 0.500 s the policeman begins to pursue the speeder with an acceleration of 6.00 m/s^2. Including the reaction time, how long does it take for the police car to catch up with the speeder?
To determine the time it takes for the police car to catch up with the speeder, we need to find the time it takes for both cars to reach the same position.
First, let's analyze the motion of the speeder. The speeder is traveling at a constant velocity of 40.0 m/s due north.
Next, let's analyze the motion of the police car. Initially, the police car is traveling at a velocity of 15.0 m/s due north. However, after a reaction time of 0.500 s, the police car starts accelerating with an acceleration of 6.00 m/s^2. So, we need to take into account the time it takes for the police car to react before it starts accelerating.
Step 1: Calculate the distance traveled by the speeder during the reaction time of the police car.
The distance traveled by the speeder during the 0.500 s reaction time can be calculated using the formula:
Distance = Velocity * Time
Distance = 40.0 m/s * 0.500 s
Distance = 20.0 m
Step 2: Calculate the relative velocity between the police car and speeder.
The relative velocity between the two cars is the difference between their velocities.
Relative Velocity = Speeder's Velocity - Police Car's Velocity
Relative Velocity = 40.0 m/s - 15.0 m/s
Relative Velocity = 25.0 m/s
Step 3: Calculate the time it takes for the police car to catch up with the speeder.
The time it takes for the police car to catch up with the speeder can be calculated using the formula:
Time = Distance / Relative Velocity
Time = 20.0 m / 25.0 m/s
Time = 0.800 s
Therefore, including the reaction time, it takes approximately 0.800 seconds for the police car to catch up with the speeder.