Answer:
It takes the putty 0.507 s to reach the ceiling.
The velocity of the putty just before it strikes the ceiling is 3.43 m/s.
Explanation:
Hi there!
The height of the glob of putty after a time "t" and its velocity are given by the following equations:
h = h0 + v0 · t + 1/2 · g · t²
v = v0 + g · t
Where:
h = height at a time "t".
h0 = initial height.
v0 = initial velocity.
t = time.
g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).
v = velocity at a time "t".
First, let´s find the time it takes the putty to reach the ceiling. We have to find at which time the putty is at a height of 3.00 m from your hand. Let´s place the origin of the frame of reference at your hand so that h0 = 0. The initial velocity, v0, is 8.40 m/s. Then:
h = h0 + v0 · t + 1/2 · g · t²
3.00 m = 0 m + 8.40 m/s · t - 1/2 · 9.8 m/s² · t²
0 = -3.00 m + 8.40 m/s · t - 4.9 m/s² · t²
Solving the quadratic equation using the quadratic formula with these coefficients:
a = -4.9
b = 8.4
c = -3
t = 0.507 s and t = 1.21 s
We use the value that happens before, that is, t = 0.507 s.
It takes the putty 0.507 s to reach the ceiling.
Now, let´s calculate the velocity of the putty at t = 0.507 s:
v = v0 + g · t
v = 8.40 m/s - 9.8 m/s² · 0.507 s
v = 3.43 m/s
The velocity of the putty just before it strikes the ceiling is 3.43 m/s.
