Tarzan, whose mass is 80.0 kg, swings from a 3.00-m vine...

Key Concepts

Inelastic Collisions

An inelastic collision is one in which the colliding bodies stick together and kinetic energy is not conserved, though momentum is. This concept is crucial for the problem because during the collision between Tarzan and Jane, some of the system's kinetic energy is lost, altering the energy dynamics and affecting the maximum height they can achieve on their upward swing.

Conservation of Momentum

In situations involving collisions, especially those that are inelastic, the total momentum of the system is conserved even though kinetic energy is not. When Tarzan picks up Jane in a perfectly inelastic collision, their combined momentum immediately after the collision is determined using the conservation of linear momentum, which then informs the energy available for their upward motion.

Conservation of Energy

This concept involves the transformation between kinetic and potential energy in a system without external work. In the context of the problem, as Tarzan swings downward, his gravitational potential energy is converted into kinetic energy, which then plays a role in determining the energy available for the subsequent upward swing after his collision with Jane.

Transcript

00:01 In this one, we're going to use a combination of the conservation of energy, as well as the conservation of momentum, to figure out at what tree limb height, what is the maximum height that tarzan and jane can get to after tarzan starts, stationary, swings down, kicks up jane, as seen by these two figures, and they're going to collide via a perfectly inelastic collision.

00:30 So they're going to both move off with the same velocity, and we're going to use both conservation momentum and energy to figure out really what's the height, the maximum height at which they can get to.

00:43 And so at first we know tarzan is at some height, and we know that he's connected to a vine that is three meters long, just shown in green.

00:54 He has a certain mass.

00:55 Jane has a certain mass.

00:57 She's on the ground stationary.

00:59 Tarzan is going to.

01:01 Swing and right before he gets to jane he is going to be moving horizontally with some tangential velocity which is in the horizontal direction when he is when the vine is perfectly vertical right before he hits jane he's going to have this velocity then they're going to undergo an inelastic collision and they're going to move off together so in order to use conservation of momentum we need to know what velocity tars in would be traveling at right before the collision.

01:38 And so we can use conservation of energy.

01:41 And i'm going to label this one because we're going to be using it twice.

01:45 So this is one and this is two.

01:52 So the initial energy of system one.

01:56 Well, tarzan is at this height.

01:59 Jane is on the ground.

02:00 And they're both stationary.

02:02 So there's no kinetic energy.

02:03 So initially it's only potential energy.

02:06 And only tarzan has them because he's above the ground.

02:09 So m sub t times g times the height of tarzan.

02:13 If we set the potential energy to be zero at the height at which tarzan would meet jane, we can set this equal to zero, then at this height the vine is three meters long.

02:31 So we can look over here.

02:32 When he hits jane, he will be three meters lower than the height at which he started with.

02:39 Since the vine over here is perfectly horizontal.

02:43 When it gets down to the point at which the vine is completely vertical, it would be the vine is three meters long, so he would have lost a height of three meters.

02:53 So we can label this as h.

02:56 That has to be his height.

02:57 He has to be three meters above where he would collide with chain.

03:02 So it will have, he will have a potential, this for a potential energy initially.

03:10 This has to equal the final.

03:11 Energy in system one, which will be the instant before he collides with jane.

03:21 So jane will still have no kinetic energy and no potential energy.

03:25 Tarzan will be at the height, which is zero potential energy.

03:31 And so we can, we know that it will, the only energy in the system will be kinetic energy of tarzan, which is one have m sub t, v sub t squared.

03:42 And since jane is on the ground, we're going to assume that tarzan will swing, will come up and be at the same height as jane so that none of them have potential energy at this final, final location.

03:58 And so we know now that, well, in this equation, we know everything but the velocity of tarzan.

04:07 So we're going to solve for that.

04:09 So if you multiply both sides by 2 and divide by m sub t find that 2g h would take the square root as well is the velocity of tarzan and we can calculate this very quick although we're going to keep a general for the momentum conservation this is 2 times 9 .81 times the height of tarzan which is 3 meters which if you plug this into a calculator the square root of 2 times 9 .81 times 3 is approximately 7 .67 meters per second.

04:50 So that's how fast tarzan will be swinging when he gets to the bottom of his arc.

04:57 When he gets to the bottom of his arc, directly afterwards, he's going to collide with chain, and they're going to move off together with some velocity v...

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