Answer:
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
