Question

An aging Hollywood actor (mass 80.0 kg) has been cloned, but the genetic replica is far from perfect. The clone has a different mass m, his stage presence is poor, and he uses foul language. The clone, serving as the actor’s stunt double, stands on the brink of a cliff 36.0 m high, next to a sturdy tree. The actor stands on top of a Humvee, 1.80 m above the level ground, holding a taut rope tied to a tree branch directly above the clone. When the director calls “action,” the actor starts from rest and swings down on the rope without friction. The actor is momentarily hidden from the camera at the bottom of the arc, where he undergoes an elastic head-on collision with the clone, sending him over the cliff. Cursing vilely, the clone falls freely into the ocean below. The actor is prosecuted for making an obscene clone fall, and you are called as an expert witness at the sensational trial. (a) Find the horizontal component $R$ of the clone's displacement as it depends on $m$ . Evaluate $R$ (b) for $m=$ 79.0 $\mathrm{kg}$ and $(\mathrm{c})$ for $m=81.0 \mathrm{kg}$ (d) What value of $m$ gives a range of 30.0 $\mathrm{m}$ ? (e) What is the maximum possible value for $R,$ and $(f)$ to what value of $m$ does it correspond? What are $(g)$ the minimum values of $R$ and (h) the corresponding value of $m ?$ (i) For the actor-clone- Earth system, is mechanical energy conserved throughout the action sequence? Is this principle sufficient to solve the problem? Explain. (j) For the same system, is momentum conserved? Explain how this principle is used. (k) What If? Show that $R$ does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

Name: Problem 56, Chapter 9: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-08-06T13:44:08-08:00
Duration: 16 min 16 s
Channel: Sheh Lit Chang

   An aging Hollywood actor (mass 80.0 kg) has been cloned, but the genetic replica is far from perfect. The clone has a different mass m, his stage presence is poor, and he uses foul language. The clone, serving as the actor’s stunt double, stands on the brink of a cliff 36.0 m high, next to a sturdy tree. The actor stands on top of a Humvee, 1.80 m above the level ground, holding a taut rope tied to a tree branch directly above the clone. When the director calls “action,” the actor starts from rest and swings down on the rope without friction. The actor is momentarily hidden from the camera at the bottom of the arc, where he undergoes an elastic head-on collision with the clone, sending him over the cliff. Cursing vilely, the clone falls freely into the ocean below. The actor is prosecuted for making an obscene clone fall, and you are called as an expert witness at the sensational trial. (a) Find the horizontal component $R$ of the clone's displacement as it depends on $m$ . Evaluate $R$ (b) for $m=$ 79.0 $\mathrm{kg}$ and $(\mathrm{c})$ for $m=81.0 \mathrm{kg}$ (d) What value of $m$ gives a range of 30.0 $\mathrm{m}$ ? (e) What is the maximum possible value for $R,$ and $(f)$ to what value of $m$ does it correspond? What are $(g)$ the minimum values of $R$ and (h) the corresponding value of $m ?$ (i) For the actor-clone- Earth system, is mechanical energy conserved throughout the action sequence? Is this principle sufficient to solve the problem? Explain. (j) For the same system, is momentum conserved? Explain how this principle is used. (k) What If? Show that $R$ does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway,… 7th Edition

Chapter 9, Problem 56

Instant Answer

Solved by Expert Sheh Lit Chang

08/06/2020

Step 1

This can be found using the equation for the speed of an object in free fall, which is $v = \sqrt{2gh}$, where $g$ is the acceleration due to gravity and $h$ is the height from which the object falls. In this case, $g = 9.8 \, m/s^2$ and $h = 1.8 \, m$, so $v = Show more…

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An aging Hollywood actor (mass 80.0 kg) has been cloned, but the genetic replica is far from perfect. The clone has a different mass m, his stage presence is poor, and he uses foul language. The clone, serving as the actor’s stunt double, stands on the brink of a cliff 36.0 m high, next to a sturdy tree. The actor stands on top of a Humvee, 1.80 m above the level ground, holding a taut rope tied to a tree branch directly above the clone. When the director calls “action,” the actor starts from rest and swings down on the rope without friction. The actor is momentarily hidden from the camera at the bottom of the arc, where he undergoes an elastic head-on collision with the clone, sending him over the cliff. Cursing vilely, the clone falls freely into the ocean below. The actor is prosecuted for making an obscene clone fall, and you are called as an expert witness at the sensational trial. (a) Find the horizontal component $R$ of the clone's displacement as it depends on $m$ . Evaluate $R$ (b) for $m=$ 79.0 $\mathrm{kg}$ and $(\mathrm{c})$ for $m=81.0 \mathrm{kg}$ (d) What value of $m$ gives a range of 30.0 $\mathrm{m}$ ? (e) What is the maximum possible value for $R,$ and $(f)$ to what value of $m$ does it correspond? What are $(g)$ the minimum values of $R$ and (h) the corresponding value of $m ?$ (i) For the actor-clone- Earth system, is mechanical energy conserved throughout the action sequence? Is this principle sufficient to solve the problem? Explain. (j) For the same system, is momentum conserved? Explain how this principle is used. (k) What If? Show that $R$ does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

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Key Concepts

Independence of Gravitational Acceleration in Range Calculation

A notable result in projectile motion analysis is that, under certain conditions, the horizontal range of a projectile can be independent of the gravitational acceleration. This occurs when the factors involving gravity in the expressions for time of flight and horizontal speed cancel out. This concept helps to understand why, despite gravity playing a role in the vertical motion, the final expression for the horizontal displacement may not include the gravitational constant, which is both mathematically elegant and conceptually unexpected.

Projectile Motion Kinematics

Once the clone is set in motion by the collision, it behaves as a projectile under the influence of gravity. The kinematics of projectile motion involves analyzing the horizontal and vertical components of the motion separately. The time of flight, determined by the vertical drop, and the horizontal velocity, imparted by the collision, combine to give the clone’s horizontal range. This approach illustrates how the displacement is calculated and how it depends on the initial conditions post-collision.

Mass Dependence in Collisions

The mass of a body plays a significant role in collision problems because it directly affects the distribution of momentum and energy. In the problem, changes in the clone’s mass alter the outcome of the collision according to conservation principles. Understanding how mass influences the collision dynamics helps in deriving the expression for the horizontal displacement as a function of the clone’s mass, as well as in determining optimum conditions for maximum or minimum ranges.

Conservation of Linear Momentum

The conservation of momentum states that in an isolated system with no external net forces, the total momentum remains constant before and after an event, such as a collision. In the context of the problem, despite the complicated motion, momentum conservation is used to analyze the elastic head-on collision between the actor and the clone. It allows one to relate the velocities and masses of the two bodies after the collision, which directly affects the subsequent projectile motion of the clone.

Conservation of Mechanical Energy

This concept refers to the idea that in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of potential and kinetic energy) in a system remains constant. In this problem, the actor’s swing from an elevated position converts gravitational potential energy into kinetic energy, setting the stage for subsequent motion. Understanding this conversion is crucial for calculating the actor’s speed at the bottom of the swing before the collision occurs.

Elastic Collision Dynamics

An elastic collision is one in which both momentum and kinetic energy are conserved. For the collision between the actor and the clone in this scenario, the fact that the collision is elastic means that the incident kinetic energy and momentum are redistributed between the two colliding bodies without loss to deformation or heat. This principle is critical for deriving the post-collision velocity of the clone, which in turn is needed to calculate the clone’s horizontal displacement.

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Transcript

00:01 This question is about a collision between an actor and his clone.

00:08 So the clone was standing near the cliff.

00:12 The cliff is 36 meter high from the ocean.

00:17 And then the clone, the actor was standing here and he's going to swing down from a height of 1 .8 meters.

00:27 Okay, this is the clone.

00:29 This is the actor.

00:30 Okay, and then after the collision, the elastic collision between the actor and the clone, the clone falls off the ocean.

00:47 So there are 11 parts in this question in part a.

00:52 So we need to find the horizontal components are of the clone's displacement as it depends on m.

01:00 So to do this, we need to find a few things because there's a collision.

01:07 So we need to find the speed of actor just before collision.

01:18 This is equal to square root 2gh.

01:22 It goes to 2 times 9 .8 times 1 .8 square root.

01:28 So this is equal to 5 .94 meters per second.

01:37 Then we want to find the speed of the clone after collision.

01:54 So this is our v2f.

02:02 So we'll be using the formula to m1 over m1 plus m2 times v1i.

02:13 So m1 is 80 kg and then v1i is 5 .94.

02:22 And then m1 plus m2 is just 80 plus m.

02:26 Okay, i'm just going to leave it as such.

02:29 Then the next thing is we want to find the time taken for the clone to fall into the ocean.

02:52 Okay, so this is just 2h prime over g and h prime is 36 meters.

03:03 Okay, and then calculate this to be 2 .71 seconds.

03:12 So the range, the horizontal components of the displacement would be, v2f times delta t so this is equal to 2 times 80 times 5 .94 divide by ad plus m times 2 .71 okay so this is equal to 2 .58 times 10 to the tree divide by ad plus m okay so this is the answer for part a.

03:54 Okay, in part b, we want to calculate r.

03:58 Here this is r, for m equals to 79 kg.

04:07 In part b, so for m, it goes to 79 kg, r is equal to 2 .58 plus 10 times 10 to the tree is divided by 80 plus 79, you get 16 .2 meters.

04:34 Okay, this is the answer for part of the 3.

04:36 Part b, next in part c, we want to repeat the same calculation.

04:50 For m equals to 81 kg, r equals to 2 .58 times 10 to the tree, divide by 80 plus 81.

05:02 We get 16 .0 meters.

05:14 In part d, what would nb? for r equals to 30 meters okay for r equals to 30 meters okay so we have 30 times 80 plus m equals to 2 .58 times 10 to the tree so m would be 2 .58 times 10 to the tree so m would be 2 .58 times 10 to the tree divided by 30 minus 80 and this is 5 .87 kg.

06:01 So this is the answer for part d.

06:13 K then in part e, what's the maximum possible value for r? so for maxrm is equal to 0 kg...

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