Suppose that a 5-foot-long chandelier falls from a 35-foot ceiling and that you are 6 feet tall and standing directly under the chandelier
(a) How long do you have to get out of the way when it drops?
=-16(t^2)+Vt+h
=-16(t^2)+0+(35-5-6) (for the bottom of the chandelier and your head)
=-16(t^2)+24
16(t^2) = 24
t^2 = 1.5
t= 1.2247
(b)If you do not duck or get out of the way, how fast will the chandelier be traveling when it hits your head?
=-32(t)+Vot
=-32(1.2247) + 0
=-39.19 ft/sec ????
yes
yes
Well, if you don't duck or get out of the way, it looks like you'll become a real-life "headbanger"! The chandelier will be traveling at a "smashing" speed of approximately 39.19 ft/sec when it meets your head. Ouch! Might want to rethink that decision.
To calculate the time you have to get out of the way when the chandelier falls, we can use the equation of motion:
h = -16(t^2) + Vt + h0
where h is the height of the chandelier, t is the time, V is the initial velocity, and h0 is the initial height.
In this case, the height of the chandelier is 35 feet and your height is 6 feet. So the equation becomes:
35 - 6 = -16(t^2) + Vt + 0
29 = -16(t^2) + Vt
Next, let's find out how fast the chandelier is traveling when it hits your head. We can use the equation:
V = -32t + Vo
where V is the final velocity, t is the time, and Vo is the initial velocity.
In this case, the chandelier falls vertically, so the final velocity V is what we are looking for, the initial velocity Vo is 0 (it starts from rest), and we know the time from part (a) is 1.2247 seconds. Substituting these values into the equation, we get:
V = -32(1.2247) + 0
V ≈ -39.19 ft/sec
Therefore, the chandelier will be traveling at approximately -39.19 ft/sec when it hits your head if you do not duck or get out of the way.
For part (a), the equation you provided,
-16(t^2) + Vt + h = 0
represents the height of the chandelier as a function of time, where h is the initial height (35 feet in this case), V is the initial velocity (0, as the chandelier is initially at rest), and t is the time in seconds.
However, you made a mistake in setting up the equation. The displacement of the chandelier from its initial height h can be calculated as -16(t^2) + Vt, so we need to set this equation equal to the total distance it has fallen, which is 35 - 5 - 6 = 24 feet.
Therefore, the correct equation should be:
-16(t^2) + Vt = 24
Now, to solve for t, we can move the terms around:
16t^2 - Vt + 24 = 0
Then, we can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 16, b = -V, and c = 24. As we have V = 0, the equation becomes:
t = (-0 ± √(0^2 - 4(16)(24))) / (2(16))
t = ± √(-4(16)(24)) / (2(16))
t = ± √(-1536) / 32
Since time cannot be negative in this context, we take the positive square root:
t = √(1536) / 32
t ≈ 1.2247 seconds
Therefore, you have approximately 1.2247 seconds to get out of the way when the chandelier falls.
For part (b), you are correct in using the equation:
-32t + V0t = 0
to calculate the velocity of the chandelier when it hits your head. However, you made a mistake in substituting the time t.
From part (a), we found that t ≈ 1.2247 seconds. Plugging in this value, the equation becomes:
-32(1.2247) + 0 = -39.19 ft/sec
So, if you do not duck or get out of the way, the chandelier will be traveling at approximately -39.19 ft/sec when it hits your head. Note that the negative sign indicates that the velocity vector is directed downwards.