A tennis ball of mass 57.0 g is held just above a basketball of mass 590 g. With their centers v

A tennis ball of mass 57.0 g is held just above a basketball...

Key Concepts

Relative Velocity in Collisions

This concept involves analyzing the speed of one object as seen from the reference frame of another. It becomes particularly important when two objects are in motion and colliding, as it assists in setting up the equations needed to apply conservation laws effectively and predict outcomes like rebound height.

Elastic Collisions

An elastic collision is one in which both momentum and kinetic energy are conserved. This concept is essential in problems where colliding bodies, such as the tennis ball and basketball, interact without a loss of kinetic energy to deformation or heat, allowing for a precise determination of rebound speeds and heights.

Gravitational Potential Energy and Free Fall

This concept involves understanding that an object's gravitational potential energy (expressed as mgh, where m is mass, g is acceleration due to gravity, and h is height) is converted into kinetic energy as it falls. It is fundamental in determining the speed of an object after it falls a certain distance without air resistance.

Conservation of Energy

The principle of conservation of energy states that energy in a closed system remains constant—transforming from one form to another. In the context of free fall, the loss of potential energy becomes the gain in kinetic energy, allowing for the calculation of velocities and other related quantities.

Conservation of Momentum

During collisions, the total momentum of the system remains constant provided there are no external forces. This principle is crucial when analyzing the interaction between the tennis ball and the basketball during their collision, as it governs the distribution of velocities post-collision.

Transcript

00:04 In this problem.

00:05 There are two malls, one is basketball and other one is tennis ball and the tennis ball is kept just over the basketball masses of both.

00:16 The balls are given legacy capital m is indicating the bounds of the basketball and the small limits indicating mass of the tennis ball which is 57 g.

00:26 And both the balls are allowed to freely fall through a distance of 1.2 m.

00:32 And then basketball is going to hit the ground.

00:35 So in first part we have to find that with what velocity the basketball will hit the ground.

00:42 Then next what will happen after this collision? so this basketball will return with certain velocity in upward direction.

00:50 Okay.

00:51 And it collides with the tennis ball which is coming in the downward direction and the basketball will be moving in the upward direction.

01:00 And this collision it was saying us to consider as elastic collision.

01:06 So after this elastic collision, the tennis ball will have certain velocity in upward direction.

01:14 So we have to find with this velocity up to what distance the tennis ball will travel.

01:20 That is what this will be the our objective to solve for the second part.

01:25 Okay, so first we are going to deal with the velocity uh or the velocity with with the basketball is going to hit the ground.

01:35 So for that we can apply energy conservation as we can see here that the mass uh basketball is falling through a distance of 1.2 m.

01:47 Let us say this distance is each one all these distances capital edge.

01:52 So from energy conservation, what we can say that if we consider this ground level to be zero potential energy level.

02:02 Okay, so here potential energy you us potential energy so this potential energy is zero.

02:14 So this is the reference line.

02:20 Now if we talk about these two positions, so whatever is the energy here.

02:25 So here energies are in two forms.

02:27 First is in a potential energy and second is kinetic energy.

02:32 And at the reference point there will be only kinetic energy as these two balls are released from rest.

02:40 It means that at this location they will not have any any kinetic energy contained in that system.

02:48 Therefore the only energy that the ball will have at this location will be the potential energy.

02:53 So basically this potential energy is going to convert it into the kinetic energy at the reference line.

03:00 So whatever is the potential energy at this location will be equal to the kinetic energy at the bottom.

03:07 So what we can write from energy conservation.

03:21 Initial potential energy.

03:29 Initial potential energy will be equal to final kinetic energy.

03:43 Okay, so what is the initial potential energy? so m.

03:46 G into capital edge.

03:49 So this m has to be capital since we are dealing for the basketball.

03:54 So this will be equals to m g.

03:59 And two edge will be equals two half mm in two we square let us say that the basketball is going to hit the ground with velocity v.

04:09 Okay, so here mass will be cancelled from both the sides.

04:14 So velocity is choir will be equals to two g h so we will be equals to square root of to g h one thing that we can observe from this expression is that this expression is independent of mass.

04:30 It means that with if the tennis ball is also falling through the same distance 1.2 m then it will also have the same velocity in the downward direction that the basketball will have.

04:43 Now what is the velocity? so we can put the value of a cheer.

04:47 Now, g.

04:48 S actual recent due to gravity which is 9.81 m per second.

04:53 The square okay into h so h is 1.2 m.

05:02 So if you saw this so velocity comes out to be 4.85 m/s.

05:09 It means that at the time of impact with the ground both the ball will be moving with a velocity 4.85 m/s.

05:18 So this is the answer for part one not in part b.

05:24 What is the situation? so the situation is before the impact the ball that is the basketball is moving with velocity there is a basketball and there is this tennis ball, okay and the basketball is moving with velocity let us say we in the downward direction and uh this enter tennis ball is also moving with the speed v.

05:56 Okay now after the impact with the ground, that velocity of the basketball changes its direction so that tennis ball is still moving with the velocity v.

06:12 In the downward direction.

06:14 Okay, but now the basketball, this is the basketball.

06:24 So it is moving with the same velocity v.

06:28 But not in upward direction.

06:30 Okay.

06:31 And these two will collide with each other with elastically means there will be elastic collision between them.

06:39 Let us say that due to this collision due to this collision, the basketball moves in upward direction with velocity v one.

06:53 And that tennis ball moves with the speed veto.

07:02 So let us say this is v.

07:03 Two and this is very even.

07:05 Okay, so this is um this is capitalism, this is a small um and this is capital.

07:11 And so here elastic collision has occurred.

07:14 So this was the situation before the elastic collision and this was the situation after the elastic collision.

07:21 And obviously we know here the value of v which is 4.85.

07:25 So this v is 4.85 m/s that we have calculated in part a so we have to find vi violent veto.

07:38 So this is going to sue the situation just before the collision.

07:52 And this is the situation just after the collision.

08:03 Okay, so as we can see here that in the school isn't there is no external forces present.

08:09 It means that momentum will be conserved.

08:12 Okay, so we can apply momentum conservation here.

08:17 So what is the initial moment? or we can write areas that is no external force.

08:33 Therefore, momentum.

08:39 Obviously we are talking about the linear momentum.

08:41 So, linear momentum will be concept.

08:50 It means that what is the uh initial linear momentum? so if we are going to consider the upward direction to be positive direction, so the velocity that the basketball will have will be the negative.

09:05 So what is the initial momentum of uh what is the initial momentum of the tennis ball? so it will be m into minus v.

09:15 Plus.

09:15 What is the momentum of the of the basketball? it will be capital um into plus we this should be equals two momentum as the both the velocities we have considered in a poor directions.

09:28 And so both of them will be positive.

09:30 So i'm into even plus capital them into.

09:34 Sorry, i have used here.

09:39 Okay, so it will be m.

09:40 V two plus capital m.

09:42 We even.

09:43 Okay, now we can put the values of here v.

09:46 And m.

09:48 So we can take a wee common here.

09:51 So we will we if we take we comment.

09:53 So it will be m capital m minus a small.

09:55 Um that will be equals two small...