A bolt is dropped from a bridge under construction, falling 96 m to the valley below the bridge. (a) How much time does it take to pass through the last 11 % of its fall? What is its speed (b) when it begins that last 11 % of its fall and (c) just before it reaches the ground?

a) It takes the bolt 0.25 s to pass the last 11% of the fall.

b) When the bolt begins to fall the last 11% of the fall its velocity is - 41.2 m/s.

c) The velocity of the bolt just before it reaches the ground is - 43.6 m/s

Explanation:

Hi there!

a) Let's calculate how much distance it is the last 11% of the fall:

96 m · 0.11 = 10.56 m

So, we have to find how much time it takes the bolt to pass from a height of 10.56 m to the ground.

First, let's calculate how much time it takes the bolt to reach a height of 10.56 m. For that we can use this equation:

h = h0 + v0 · t + 1/2 · g · t²

Where:

h = height of the bolt at a time t.

h0 = initial height.

v0 = initial velocity.

t = time.

g = acceleration due to gravity.

If we consider the ground as the origin of the frame of reference, then h0 = 96 m. Since the bolt is dropped, the initial velocity is zero (v0 = 0). Then, the equation gets reduce to this:

h = h0 + 1/2 · g · t²

We have to find at which time h = 10.56 m.

10.56 m = 96 m - 1/2 · 9.8 m/s² · t²

Solving for t:

√ (-2 · (10.56 m - 96 m) / 9.8 m/s²) = t

t = 4.2 s

Now that we have the time at which the bolt is located at 10.56 m above the ground, we can calculate the velocity of the bolt at that time.

The equation of velocity (v) of the bolt is the following:

v = v0 + g · t

at t = 4.2 s.

v = 0 - 9.8 m/s² · 4.2 s

v = - 41.2 m/s

When the bolt begins to fall the last 11% of the fall its velocity is - 41.2 m/s.

Now, we can calculate how much time it takes to fall the last 10.56 m.

The initial velocity of the bolt will be the velocity at h = 10.56 m. The initial height will be 10.56 m.

h = h0 + v0 · t + 1/2 · g · t²

We have to find the time at which h = 0 (the bolt hits the ground)

0 = 10.56 m - 41.2 m/s · t - 1/2 · 9.8 m/s² · t²

Solving the quadratic equation using the quadratic formula:

t = 0.25 s (the other solution of the quadratic equation is negative and thus discarded).

It takes the bolt 0.25 s to pass the last 11% of the fall.

Now, let's calculate the velocity of the bolt when it reaches the ground:

v = v0 + g · t

v = - 41.2 m/s - 9.8 m/s² · 0.25 s

v = - 43.6 m/s

The velocity of the bolt just before it reaches the bolt is - 43.6 m/s