# Vectors of Freely Falling Bodies

scorpa
Ok guys, here is a question I have about freely falling bodies. I did do the questions but my answer just doesn't seem right to me, I don't think I went about it the right way. I will show what I did below the question:

Determined to test the law of gravity for himself, a student walks off a skyscraper 180m high, stopwatch in hand, and starts his freefall (zero initial velocity). Five seconds later, superman arrives at the scene and dives off the roof to save the student.

A) Superman leaves the roof with an inital speed Vo, that he produces by pushing himself downward from the edge of the roof with his legs of steel. He then falls with the same acceleration as any freely falling body. What must his initial velocity be so that he reaches the student just before he hits the ground?

B) On the same graph sketch the position of the student and of Superman as functions of time. Take Superman's initial speed that was calculated in part a.

C) If the height of the skyscraper is less than some minimum value, even Superman can't reach the student before he hits the groun What is this minimum height?

My Solution:

a) First I found the time it would take for the student to fall 180 m. Using the equation x(t)=Xo+VoT+(1/2)at^2, I found this time to be 6.058s after I plugged in the numbers. It then says that Superman doesn't reach the scene until 5s after the student jumps, which would mean that he only has 1.058s left to save the student. So I then plugged those numbers into the same equation listed above and got an initial velocity of 31.0m/s downwards. It made sense to me at the time, but I am starting to doubt myself.

b) I was going to wait to draw that graph until I was sure of how to do a.

c) I'm not really sure how to do part c. I was thinking of making the two position functions equal to each other and solving for the height but I don't think that will work after all.

Thanks for any advice you can give.

## Answers and Replies

shaner-baner
Think about it. If the student hits the ground before 5 seconds, superman can't save him no matter how fast he goes. I'm not going to say anymore

scorpa
Oh, ok I think that helps a lot, I feel stupid now!

Do you think method I used to get my first answer looked correct?

Homework Helper
scorpa said:
Oh, ok I think that helps a lot, I feel stupid now!

Do you think method I used to get my first answer looked correct?
You method is ok and your time of 6.058s is correct, but superman's velocity of 31m/s is wrong.
Superman's got about 1 sec to reach the bottom of the skyscraper. Acceleration due to gravity will increase his velocity in that time by about 10 m/s giving him a final velocity of 40 m/s and thus an average velocity of 35 m/s, say. And he's not going to cover 180 m in 1 sec at 35 m/s!. So, just by looking/checking you can see that your answer is wrong.

Since he wants to cover 180 m in just over 1 sec then his average velocity should be a bit less that 180 m/s, say 175 m/s.

scorpa
I know that he won't be able to rescue the student by going only 31m/s is wrong, I was saying that his initial velocity when he pushed off the building is 31m/s, from there he accelerates downward and his velocity will increase, but initally it is 31m/s. Is this still wrong?

scorpa
****bump****

Homework Helper
scorpa said:
I know that he won't be able to rescue the student by going only 31m/s is wrong, I was saying that his initial velocity when he pushed off the building is 31m/s, from there he accelerates downward and his velocity will increase, but initally it is 31m/s. Is this still wrong?
Still wrong, I'm afraid.
He then falls with the same acceleration as any freely falling body. What must his initial velocity be so that he reaches the student just before he hits the ground?
Since he falls with the acceleration of gravity, then that is only about 10 m/s every second. And since he only has about 1 sec then his final velocity will be only about 40 m/s. etc
So his initial velocity must much greater.

scorpa
Ok, now I see what you were saying. I'm not really sure how to get a higher initial velocity, I understand why you say that it needs to be close to 180m/s, but how do I determine that mathematically?

Homework Helper
scorpa said:
Ok, now I see what you were saying. I'm not really sure how to get a higher initial velocity, I understand why you say that it needs to be close to 180m/s, but how do I determine that mathematically?
A) Superman leaves the roof with an inital speed Vo,
The problem translates to.

Question: An object falls from the top of a skyscraper, 180m tall, with an initial downwards speed Vo and falls freely under gravity. It hits the ground 1.058 s later.
What is the initial velovcity?

Can you solve that question ?

scorpa
-180m=v(1.058s) +(0.5)(9.81)(1.058)^2
164.9m/s is the initial velocity required.

That makes more sense! Thanks!

scorpa
The weird thing is that is exactly what I had done the first time, I must have gotten a number entered into my calculator wrong.

scorpa
oops that's still wrong the answer is actually 174.976m/s...darn my fumbling fingers!

Homework Helper
Oh dear scorpa, you're all over the place

The inti velocity is actually Vo = 164.94 m/s

180 = Vo(1.058) + 0.5*9.81*(1.058)²
180 = 1.058*Vo + 5.49
1.058*Vo = 180 - 5.49 = 174.51
Vo = 174.51/1.058
Vo = 164.94 m/s
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