While repairing a defective radio transmitter located 410 m up on the Skydeck
of Chicago’s Sears Tower, Lyle drops his hammer that falls all the way to the
ground below. a) How long will it take for Lyle’s hammer to fall? b) With what
speed will the hammer hit the pavement? c) How far will the hammer have
fallen after 1.50 s when a janitor watches it pass outside an office window?
To answer these questions, we can use the equations of motion to find the time, speed, and distance of the hammer's fall. The key equation we will use is:
h(t) = h_0 - (1/2) * g * t^2
Where:
- h(t) is the height of the hammer at time t
- h_0 is the initial height of the hammer (410 m in this case)
- g is the acceleration due to gravity (approximately 9.8 m/s^2) which acts downward
- t is the time elapsed
Now let's answer each question step by step:
a) How long will it take for Lyle’s hammer to fall?
We need to find the time it takes for the hammer to fall from the Skydeck to the ground. Since we know the initial height (h_0) is 410 m and the acceleration due to gravity (g) is -9.8 m/s^2 (negative because it acts downward), we can substitute these values into the equation:
0 = 410 - (1/2) * 9.8 * t^2
Simplifying the equation, we get:
4.9t^2 = 410
Now we can solve for t by dividing both sides of the equation by 4.9 and taking the square root:
t = sqrt(410/4.9)
Using a calculator, we find that t is approximately 9.05 seconds.
Therefore, it will take Lyle's hammer approximately 9.05 seconds to fall.
b) With what speed will the hammer hit the pavement?
To find the speed of the hammer when it hits the pavement, we can use the equation for velocity:
v(t) = -g * t
Substituting the value of g and the time t that we found earlier, we get:
v = -9.8 * 9.05
Calculating this value, we find that the hammer will hit the pavement with a speed of approximately 88.79 m/s.
c) How far will the hammer have fallen after 1.50 s when a janitor watches it pass outside an office window?
To find the distance the hammer will have fallen after 1.50 seconds, we can use the equation for height:
h(t) = h_0 - (1/2) * g * t^2
Substituting the values, we get:
h = 410 - (1/2) * 9.8 * (1.5)^2
Calculating this expression, we find that the hammer will have fallen approximately 448.57 m after 1.50 seconds.
Please note that in reality, there might be some air resistance affecting the hammer's fall, but for the purpose of this calculation, we have assumed that it is negligible.