Question

Atman was driving the batmobile at 105 mph (=154 ft/sec), when he sees a brick wall directly ahead. When the batmobile is 500 ft from the wall, he slams on the brakes, decelerating at a constant rate of 24 ft/s². Does he stop before he hits the brick wall? If so, how many feet to spare? If not, what is his impact speed?

Answer 1

Using the kinematic equation to account for acceleration, Batman will stop 4.25 feet before hitting the brick wall when decelerating at a rate of 24 ft/s² from an initial speed of 154 ft/s over a distance of 500 feet.

Step-by-step explanation:

To determine if Batman will stop before he hits the brick wall, we can use the kinematic equation Δx = ν0t + ½at2, where Δx is the displacement (distance until the wall), ν0 is the initial speed, t is the time, and a is the acceleration. Given that Batman's initial speed is 154 ft/s and he decelerates at 24 ft/s2, we need to find the time it takes for the batmobile to come to a stop by using the equation ν = ν0 + at, and solve for t by setting ν to 0.

The calculation would look like this:

0 = 154 ft/s - (24 ft/s2 * t)

t = 154 ft/s / 24 ft/s2 = 6.4167 seconds

Now, we substitute t back into the displacement formula to find Δx:

Δx = 154 ft/s * 6.4167 s + ½(-24 ft/s2)*(6.4167 s)2

Δx = 987.5 ft - 491.75 ft

Δx = 495.75 ft

Since he was initially 500 ft from the wall and only traveled 495.75 ft, he would stop 4.25 feet from the wall.