ssm Starting with an initial speed of 5.00 m / s at a height of 0.300 m, a 1.50-kg ball swings d

Ssm Starting with an initial speed of 5.00 m / s at a height...

Key Concepts

Conservation of Mechanical Energy

This principle states that in a closed system where non-conservative forces (like friction) are negligible, the total mechanical energy—comprising kinetic and potential energy—remains constant. It is used to relate the energy at one point of an object's motion to the energy at another, allowing for the calculation of speed changes due to variations in height or potential energy.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. This concept is vital for solving problems where two objects collide without any loss of energy to sound, heat, or deformation, enabling the determination of the final velocities of the objects involved.

Conservation of Momentum

In any collision occurring in the absence of external forces, the total momentum of the system remains unchanged. This key concept is used to set up equations that relate the masses and velocities of colliding objects before and after the collision, providing a means to solve for unknown velocity values.

Transcript

00:01 We have a 1 .5 kilogram ball in the end of a string, which is already moving with the speed of 5 meters per second.

00:11 And it swings down from the string and it hits, i don't think it's quite at such an angle, something like that.

00:20 And it hits another ball.

00:24 And the ball is sitting there and they have an elastic collision.

00:27 So there's multiple parts of the problem.

00:29 First, we need to know what is the speed of the first ball just before it collides with the stationary ball.

00:35 So for this part, we can use conservation of energy.

00:38 This is a simple pendulum.

00:40 We know that at the top, the ball has both potential energy and kinetic energy.

00:48 And at the bottom, as long as we define our reference frame, our zero to be the lowest point of the swing, then it only has kinetic energy.

01:00 So we know that energy is conserved, so we can say that the potential energy at the top plus the kinetic energy at the top is equal to the kinetic energy at the bottom.

01:14 So we know the formula for potential energy is mg times the height.

01:19 So, oh, and i did forget to tell you that the height that we're starting with is 0 .3 meters.

01:25 So we have a mass of 1 .5 times 9 .8.

01:35 Times the height, which is 0 .3, plus one -half times the mass times the velocity squared, is equal to the kinetic energy at the bottom.

01:55 One -half times the mass times velocity at the bottom, which we don't know, squared.

02:03 So you can see that the 1 .5s cancel out.

02:05 We could have done that before plugging out our numbers.

02:08 It's a little bit, it looks messy.

02:10 It's not so bad, but this is 9 .8 times 0 .3 plus a half of 0 .5 squared is going to be equal to a half of v squared.

02:18 So if you solve that using your basic algebra, you get v at the bottom of 5 .56 meters per second.

02:29 So that's the answer to part a.

02:31 We're done with the first part of the question.

02:34 But then the next part of the question says, what are the velocities of both balls after impact.

02:40 They tell us that the collision is elastic.

02:42 And this is one of those cases where we have an elastic collision where the initial velocity of the second object is zero.

02:54 So v .0 object 2 is zero.

02:57 So in the text, which i've referred to this in other problems, there is a shortcut.

03:05 It's not an that you can use for any other situation.

03:10 This is just for when the initial velocity is zero for the second object.

03:15 But the relationship between the final velocity of the first object and the initial velocity of the first object is this.

03:24 It's the v final one, is the difference of the two masses divided by the sum of the two masses times the initial velocity of object one.

03:35 So this can help us because we're going to use conservation of momentum, but without this extra relationship, we would find that we didn't have enough knowns.

03:44 We had too many unknowns.

03:46 So let's save that for a second, and we'll do conservation of energy.

03:50 So i'm sorry, conservation of momentum.

03:52 The initial momentum is equal to the final.

03:55 In this case, what's initially moving, we have the first ball, which strikes the second ball, and the second ball is at rest.

04:05 So that one is going to be zero.

04:07 That's going to to be equal to the mass of the first ball times the v final of the first ball times the mass of the second ball times the final of the second ball so we do have some values we can plug some numbers in but we're going to get stuck because we don't know enough so let's start with that we'll say mass one is 1 .5 and 5 .56 was our velocity just before impact and that's going to be equal to 1 .5 times v -final 1 plus 4 .6, which is the mass of our second ball, times v -final 2.

04:53 So we have two unknowns, and that doesn't work.

04:56 So let's use the special relationship here to find the final one.

05:03 I'll keep it in sight, but write it down here.

05:07 So v -final 1 is going to be the difference of the masses...

Please give Ace some feedback