00:01
In this question, we have the earth, and we have a ball not necessarily drawn to scale.
00:08
I know the mass of the ball.
00:10
500 grams.
00:12
I'm going to put that in kilograms, which is important.
00:16
And the mass of the earth, hopefully it's located in the back cover of your book, but 5 .98 times 10 to the 24th kilograms.
00:31
And the question is, if i drop this from a height of, let me draw that height a little bit better, a height of 10 meters, and it has a perfectly elastic collision with the earth.
00:44
What's the earth's velocity after the collision, which is just really, really fascinating? this is actually a two -part problem because we need to use conservation of energy to figure out the speed of the ball just before impact.
01:01
Impact and then we use momentum and the elastic collision equation to figure out the speed of the earth.
01:11
So let's go ahead and do that first part.
01:15
Ei is equal to ef.
01:17
It's conservation of energy.
01:19
I start with purely potential energy, assuming the service of the earth is my arbitrary zero, and it's turned entirely into kinetic energy, one -half mv squared.
01:31
Solving for v, v is equal to the square root of 2gh.
01:37
And so that's going to be equal to the square root of 2 times 9 .8 times 10.
01:48
And so let's just plug to that in to the calculator, square root of 196.
01:59
And that's going to be 14 meters per second.
02:02
And so that is the speed of the collision.
02:10
We're assuming the earth itself has a velocity of zero.
02:13
And so i've got two equations that i need to use.
02:18
I'm going to write the first one here.
02:21
M1 v1 plus m2 v2 is equal to m1 v1 prime plus m2 v2 prime.
02:29
It's just conservation momentum.
02:31
The momentum of each of my two objects before collision is equal to the momentum of the two objects after the collision.
02:38
I'm going to write the equation for the elastic collision over here because i want to be able to use that equation on the right.
02:48
It is just that v1 minus v2 is equal to v2 prime minus v1 prime.
02:56
So the difference in the velocities before the collision is equal to the difference in the velocities after the collisions by the first one i have v1 as my starting point.
03:06
Second collision, after the collision, i have v2.
03:09
And that is just an excellent equation to use.
03:13
I've got two equations, two unknowns.
03:15
So if i use this equation and this equation, i should be able to solve for whichever i want.
03:21
Now, i'm going to go ahead and make v2 represent the earth.
03:25
I know that i'm starting off with a zero velocity, by the way.
03:29
But i want to solve for v2, so i need v1 to go away.
03:34
And so if i write this equation here, v1 minus v2 is equal to v2 prime minus v1 prime.
03:46
As long as i can put an m1 in front of that, then i'll be able to add those equations and v1 will go away and i'm left with is v2.
03:56
So i'm going to multiply the entire equation by m1.
04:01
I'm going to put the minus side over here...