Suppose that a senior driving a Pontiac zooms out of a darkened tunnel at 31.0 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 15.7 m/s. She hits the brakes as fast as she can (her reaction time is 0.40 s). If she can decelerate at 2.5 m/s2, what is the minimum distance between the driver and the bus when she first sees it so that they do not collide?
89 m
To solve this problem, we can use kinematic equations to calculate the minimum distance between the driver and the bus.
First, we need to find the time it takes for the driver to react. From the information given, we know that the reaction time is 0.40 seconds.
Next, we can find the distance covered during the reaction time. We can use the equation:
distance = initial velocity * time
The initial velocity is the speed of the senior driving out of the tunnel, which is given as 31.0 m/s. The time is the reaction time, which is 0.40 seconds.
distance = 31.0 m/s * 0.40 s = 12.4 m
Now, let's find the time it takes for the driver to slow down and match the speed of the bus. We can use the equation:
time = (final velocity - initial velocity) / acceleration
The final velocity is the speed of the bus, which is 15.7 m/s. The initial velocity is the speed of the senior driving out of the tunnel, which is 31.0 m/s. The acceleration is given as -2.5 m/s^2 (negative because it represents deceleration).
time = (15.7 m/s - 31.0 m/s) / -2.5 m/s^2 = 6.12 s
Finally, we can find the distance the senior driver travels during this time using the equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
Substituting the values:
distance = 31.0 m/s * 6.12 s + 0.5 * (-2.5 m/s^2) * (6.12 s)^2 = 116.4 m
Therefore, the minimum distance between the driver and the bus when she first sees it so that they do not collide is the sum of the distance covered during the reaction time and the distance covered while decelerating:
minimum distance = 12.4 m + 116.4 m = 128.8 m