Question

Consider a railroad bridge over a highway. A train passing over the bridge dislodges a loose bolt from the bridge, which proceeds to fall straight down and ends up breaking the windshield of a car passing under the bridge. The car was 27 m away from the point of impact when the bolt began to fall down; unfortunately, the driver did not notice it and proceeded at constant speed of 17 m/s. How high is the bridge? Or more precisely, how high are the railroad tracks above the windshield height? The acceleration due to gravity is 9.8 m/s 2 .

Answer 1

By using the constant speed of the car and the time it took for it to reach the point under the bolt, we calculated the time of fall to be 1.588 seconds. Then, given the acceleration due to gravity, we determined that the height of the bridge above the windshield is approximately 12.35 meters.

Step-by-step explanation:

To find the height of the bridge above the windshield, we first need to calculate the time it took for the bolt to hit the car. Since the car was moving at a constant speed of 17 m/s and it was 27 m away from the point right below where the bolt fell, we can calculate the time it took for the car to reach that point:

Time = Distance / Speed

= 27 m / 17 m/s

= 1.588 seconds

Now, we can use this time to calculate how far the bolt fell. The bolt was in free fall, so we can use the equation for the distance traveled under constant acceleration due to gravity:

Height = 0.5 * g * t²

Substitute g (acceleration due to gravity) with 9.8 m/s² and t with the time calculated:

Height = 0.5 * 9.8 m/s² * (1.588 s)²

Height = 0.5 * 9.8 m/s² * 2.5211 s²

Height = 12.353 m

Therefore, the height of the railroad tracks above the windshield height is approximately 12.35 meters.

Answer 2

The railroad tracks are 13 m above the windshield (12 m without intermediate rounding).

Step-by-step explanation:

First, let´s calculate the time it took the driver to travel the 27 m to the point of impact.

The equation for the position of the car is:

x = v · t

Where

x = position at time t

v = velocity

t = time

x = v · t

27 m = 17 m/s · t

27 m / 17 m/s = t

t = 1.6 s

Now let´s calculate the distance traveled by the bolt in that time. Let´s place the origin of the frame of reference at the height of the windshield:

The position of the bolt will be:

y = y0 + 1/2 · g · t²

Where

y = height of the bolt at time t

y0 = initial height of the bolt

g = acceleration due to gravity

t = time

Since the origin of the frame of reference is located at the windshield, at time 1.6 s the height of the bolt will be 0 m (impact on the windshield). Then, we can calculate the initial height of the bolt which is the height of the railroad tracks above the windshield:

y = y0 + 1/2 · g · t²

0 = y0 -1/2 · 9.8 m/s² · (1.6 s)²

y0 = 13 m