Suppose that a woman driving a Mercedes zooms out of a darkened tunnel at 33 m/s. She is momentarily blinded by the sunshine. When she recovers, she sees that she is fast overtaking a bus ahead in her lane moving at the slower speed of 16 m/s. She hits the brakes as fast as she can but her reaction time is 0.3 s.
If she can decelerate at 2.1 m/s2, what is the minimum distance between the driver and the bus when she first sees it so that they do not collide?
To solve this problem, we need to consider two main factors: the distance traveled during the driver's reaction time and the distance needed to bring the car to a stop.
Let's break down the problem step by step:
1. Distance traveled during the driver's reaction time:
The driver's reaction time is given as 0.3 seconds. During this time, the car is still moving at its initial velocity. To find the distance traveled during this time, we use the formula:
Distance = Velocity * Time
Distance = 33 m/s * 0.3 s = 9.9 meters
2. Distance needed to bring the car to a stop:
The car is traveling at 33 m/s, and it needs to decelerate (negative acceleration) to a stop. The deceleration is given as 2.1 m/s^2. To find the distance needed to stop, we use the formula:
Distance = (Velocity^2 - Final Velocity^2) / (2 * Acceleration)
Final Velocity is 0 m/s (since the car needs to stop)
Distance = (33 m/s)^2 / (2 * -2.1 m/s^2) = 283.5 meters
3. Total minimum distance between the driver and the bus:
To find the minimum distance, we add the distance traveled during the driver's reaction time to the distance needed to bring the car to a stop:
Total Distance = Distance during Reaction Time + Distance needed to Stop
Total Distance = 9.9 meters + 283.5 meters = 293.4 meters
Therefore, the minimum distance between the driver and the bus when she first sees it so that they do not collide is 293.4 meters.