Question

PLEASE HELP!

Jack and John are swingin on two different swing seats, but their max heights are about the same. Knowing that Jack is half the mass of John...

A) How will their velocities compare at the bottom of the swing?
B) will they arrive at the bottom of the swing at the same time?

Answer 1

Johns velocity will be half of Jacks?

no they won't?

Answer 2

You need to apply the law of conservation of energy. Let's make an assumption that jack has a mass of 50 kg and John a mass of 100 kg to prove our point. If we solve the problem for velocity for jack then john we can make a comparison between the two and see what we really know. I'll make an equation next showing this.

Answer 3

\[Ke = Gpe\]
\[\frac{ 1 }{ 2 }mv ^{2} = mgh\]
\[\frac{ 1 }{ 2 }v ^{2} = gh\]
\[v ^{2} = 2gh\]
\[v = \sqrt{2gh}\]
If we follow the above progression, we first set the kinetic energy of the person at the bottom of the swing equal to the gravitational potential energy at the top of the swing. Since both sides have mass in common to the other, it mathematically crosses out. Even if we plugged in, say 50 kg, for the mass in the problem, 50/50 = 1 so what the math is trying to say is that the mass has NO BEARING on the velocity of a person swinging. This being said, if both persons were swinging from the same height, they would both have the same velocity.

Answer 4

Their period of oscillation should also be independent of the mass involved, thus they would do everything identical. This is because the equation for the period of a pendulum is \[T = 2 \pi \sqrt{\frac{ l }{ g }}\]
Since mass is not a part of this equation, is has absolutely no bearing on how a person swinging (which is like a pendulum) will influence the time. Again, if there is no mass in the equation that involves period/time then they do not depend on one another so mass will NOT influence either velocity or time. Does this make sense?

Answer 5

wow

Answer 6

im not gunna lie, thats hard to follow :P Half those equations we never even learned lol, we only learned Kinetic Energy and simple equations like that :P

Answer 7

lets say if i plug in a random number for Kinetic energy in Ek equation so...:
800=1/2(10)v^2 then do
800=1/2(5)v^2

the guy with the smaller mass will have the higher velocity...would that not work?

Answer 8

no matter what kinetic energy i put in the smaller mass will always have the higher velocity.is that not right?

Answer 9

Intuition would tell us to expect that however it is not the case. Have you discussed the conservation of energy before and do you understand what that means? Perhaps I can help you through it in another way.

Answer 10

i have heard of it but i am a little fuzzy on the topic :P?

Answer 11

Ok so if you hold a ball 1 meter above the ground, what kind of energy does it have?

Answer 12

Gpe?

Answer 13

since gravity is pulling it down?

Answer 14

yes and why?

Answer 15

excelelnt!

Answer 16

When you let it go, what happens?

Answer 17

it falls
to the ground then boucnes

Answer 18

because the force dropping it will cause it to bounce...how high is dependant on how heavy the ball is?

Answer 19

@Scienceguy would that be correct? does that mean mass is important?

Answer 20

Mass is required to calculate GPE. The fact a ball bounces has to deal with its coefficient of restitution. BUT my question to you is that when the ball is falling, as its falling toward the ground what kind of energy does it now have?

Answer 21

:S i don't know, gravitational? and I have never heard about coefficient of restitution before .. sorry :S

Answer 22

I mentioned restitution because you said about a bounce... Ignore that.... Yes the ball does have gravitational potential energy because it is still elevated above the ground but it must also have kinetic energy (motion energy) because it is actively moving. Does that make sense so far?

Answer 23

yes it does

Answer 24

Ok. So the energy was initially purely gravitational and now its being converted into some form of kinetic energy. This is what the conservation of energy says. Energy is not created or destroyed, only transformed from one energy form to another.

This is how we can approach the problem. There are 3 types of mechanical energy: Kinetic energy, gravitational potential energy, and elastic potential energy. When either person is at their highest point in their swing, we need to ask ourselves 3 questions:
"Is it moving (KE)? Is it elevated (GPE)? Is it on a spring (EPE)? So when you or anyone is swinging on a swing, and you are at that exact instant in which you are as high above the ground as you can be, which kind of energy do you suppose you have?

Answer 25

GPE?

Answer 26

YES!!! Excellent!!!

Answer 27

This is the only kind of energy present in that case as you are not moving because you are at the highest point and stopped for a split second, you are elevated so you correctly mention the GPE, and you are not on a spring so there is no EPE. Good.

What about when you are at the bottom of your swing? Are you moving? Can you go any lower than you are? (GPE) Are you on a spring? What kind of energy is only present at the bottom of a swing?

Answer 28

Kinetic energy

Answer 29

YES!!

Answer 30

So the GPE of John = KE of John. His energy was converted from GPE to KE. So that is why I made that huge equation at the top. KE = GPE for John. With me there so far?

Answer 31

yee

Answer 32

Ok. What is the equation to determine GPE? Lets pretend John has a mass of 100 kg and lets also use g = 10 m/s^2 for the acceleration of gravity to keep things simple. If John swings to a maximum height of 1 m above the ground on either side of the swing, what is his GPE for this example case?

Answer 33

Any luck?

Answer 34

1000?

Answer 35

mgh , (100)(10)(1)

Answer 36

Yep

Answer 37

BUT... you also said that GPE at the top = KE at the bottom correct??

Answer 38

so both john and jack's mgh are =?

and ye i did say that

Answer 39

Dont worry about Jack just yet. You'll see where I'm going.
So the equation for KE is KE = 1/2 mv^2. We want to know how fast John is going. We said his original mass was 100 kg and we said that the KE = GPE = 1000 J. If we rearrange the equation for velocity it should look like this:
\[v=\sqrt{\frac{ 2KE }{ m }}\]

What I would like you to do is plug in 1000 for KE, 100 for m (b/c these are the numbers we used) follow the equation and tell me what your velocity for John is.

Answer 40

14.1?

Answer 41

Here's what I'd like you to try. Take 2 x 1000 and divide by 100 and tell me what number you get. (I think I know what mistake you made). We'll then go from there.

Answer 42

20

Answer 43

20 squarerooted is 4.47

Answer 44

Yep. Good. That is how fast John would be moving at the bottom. 4.47 m/s. Lets consider jack now. Jack has half the mass of john so what is jack's mass based off our numbers?

Answer 45

50?

Answer 46

Yep. So I want you to calculate his initial GPE if he was also 1 meter above the ground, mass of 50 kg, and still use 10 for g. Tell me what number you get.

Answer 47

500

Answer 48

Excellent. Lets do the same again. We know that jack is moving the fastest at the bottom as well so what is his kinetic energy at the bottom if his Grav. Energy is 500 J at the top?

Answer 49

since we said its = its 500?

Answer 50

Very well done. So we know that KE = 500 and m = 50 (jacks mass). Lets use the same equation as above. I'll rearrange it for v. I want you to plug in and see what you get. Remember, do the sqrt last.
\[v =\sqrt{\frac{ 2KE }{ m }}\]

Answer 51

4.47!!

Answer 52

wow man, your the best teacher

Answer 53

Exactly. So what did we learn from this? What impact does mass have on the velocity at the bottom of a swing?

Answer 54

well mass has no bearing

Answer 55

even though he was heavier they ended up being equel

Answer 56

there velocity that is

Answer 57

you are 100% correct. Do me a quick favor and scroll up towards the start of our conversation. Take a look at that huge mess of a series of equations I showed you that solved for v. Does it make any moer sense to you now? I'm just curious.

Answer 58

wow yes it does

Answer 59

does this entire mgh=ke have the same to do with b?

Answer 60

since their velocities = does that mean they will leave the swing going the same velocity and since gravity being constant pull them down at teh same time?

Answer 61

or are their other factors?

Answer 62

maybe when they leave the swing?

Answer 63

yes... and if you also consider that because they leave the same height, they must also pass through the same distance with the same velocity so they would reach there at the same time. Its like driving down a road in a car beside another car that is accelerating at the same rate with the same initial velocity.

As long as they are on the swing, this math will always apply. Once they leave the swing, now our math model may need some tweaking but for this instance, their times should be the same as well.

Answer 64

wow, in 3 years of learning this stuff, i have never been taught so well. Great job man, I hope to be helped by you in the future :P

Answer 65

Not a problem sir(??). I'm a physics/math teacher for a living and do this when I'm bored. Keep an eye out and dont hesitate to ask me a question.

Answer 66

thanks again! Your one heck of a teacher!

Answer 67

You're too kind. Thank you and good luck.