Suppose that a senior driving a Pontiac zooms out of a darkened tunnel at 39.4 m/s. She is momentarily blinded by the sunshine. When she recovers, she sees that she is fast overtaking a camper ahead in her lane moving at the slower speed of 12.9 m/s. She hits the brakes as fast as she can (her reaction time is 0.36 s). If she can decelerate at 3.6 m/s^2, what is the minimum distance between the driver and the camper when she first sees it so that they do not collide?
To find the minimum distance between the driver and the camper so that they do not collide, we need to consider the distance covered by the driver during her reaction time and the distance covered while decelerating.
Let's begin by calculating the distance covered by the driver during her reaction time:
Distance covered during reaction time = Initial velocity × Reaction time
Given:
Initial velocity = 39.4 m/s
Reaction time = 0.36 s
Distance covered during reaction time = 39.4 m/s × 0.36 s
= 14.184 m
Now, let's calculate the distance covered while decelerating:
To stop the vehicle, the driver needs to reduce its speed from 39.4 m/s to the slower speed of the camper, which is 12.9 m/s. The change in velocity is:
Change in velocity = Initial velocity - Final velocity
= 39.4 m/s - 12.9 m/s
= 26.5 m/s
Using the equation of motion:
Distance covered = (Change in velocity)^2 / (2 × Acceleration)
Given:
Change in velocity = 26.5 m/s
Acceleration = -3.6 m/s^2 (negative because the car is decelerating)
Distance covered while decelerating = (26.5 m/s)^2 / (2 × -3.6 m/s^2)
= 356.597 m
Therefore, the total distance covered by the driver before coming to a stop is:
Total distance = Distance covered during reaction time + Distance covered while decelerating
= 14.184 m + 356.597 m
= 370.78 m
Hence, the minimum distance between the driver and the camper when she first sees it so that they do not collide is approximately 370.78 meters.
To find the minimum distance between the driver and the camper when she first sees it so that they do not collide, we can break down the problem into three parts: the distance traveled during the driver's reaction time, the distance traveled while decelerating, and the distance traveled by the camper.
1. Distance traveled during the reaction time:
During the driver's reaction time, the Pontiac continues moving at its initial speed. The distance traveled during this time can be calculated using the formula:
Distance = Speed x Time
Given that the speed of the Pontiac is 39.4 m/s and the reaction time is 0.36 s, we can calculate the distance traveled during the reaction time:
Distance = 39.4 m/s x 0.36 s = 14.184 m
2. Distance traveled while decelerating:
Once the driver hits the brakes, the Pontiac starts to decelerate at a rate of 3.6 m/s^2. To calculate the distance traveled while decelerating, we can use the following formula:
Distance = (Initial Speed)^2 / (2 x Deceleration)
The initial speed is 39.4 m/s, and the deceleration is 3.6 m/s^2, so we can calculate the distance:
Distance = (39.4 m/s)^2 / (2 x 3.6 m/s^2) = 428.75 m
3. Distance traveled by the camper:
While the driver is reacting and decelerating, the camper is also moving. The distance traveled by the camper can be calculated by multiplying its speed by the total reaction time, including the time spent decelerating:
Distance = Speed x Total Time
Given that the speed of the camper is 12.9 m/s and the total time is the driver's reaction time plus the time spent decelerating, we can calculate the distance traveled by the camper:
Distance = 12.9 m/s x (0.36 s + (39.4 m/s / 3.6 m/s^2)) = 67.384 m
Now, to find the minimum distance between the driver and the camper when she first sees it so that they do not collide, we need to subtract the sum of the distances traveled during the reaction time and while decelerating from the distance traveled by the camper:
Minimum Distance = Distance traveled by the camper - (Distance traveled during reaction time + Distance traveled while decelerating)
Minimum Distance = 67.384 m - (14.184 m + 428.75 m)
Minimum Distance = -375.55 m
The negative value means that the driver has already passed the camper, and they would have collided. Therefore, in this scenario, a collision is unavoidable.
s₀=v₀•t₀=39.4•0.36 =14.2 m
s1=v²-v₀²)/2(-a) = (2.9²-39.4²)/2(-3.6)=5.17 m.
s= s1+s₀=5.17+14.2=19.37 m.