A truck with mass m has a brake failure while going down an...

Transcript

00:01 Hello, everyone.

00:03 And in this video, i'm going to be solving problem 58 from chapter 7 of the sears and zamansky's university physics textbook.

00:11 So the first thing i'm going to do is i'm going to read through the problem, make sure that we fully understand the problem and what the problem is asking of us and what the problem expects us to actually do.

00:23 All right.

00:25 So problem 58.

00:27 A truck with mass m has a break failure while going down an icy mountain road.

00:32 Of constant downward slope angle alpha.

00:35 And that's shown in figure p -758.

00:38 And that's the same figure that i have crudely drawn already on screen.

00:44 Initially, the truck is moving downhill at speed v -0.

00:48 After careening downhill at distance l with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle beta.

00:58 The truck ramp has a soft sand surface for which the coefficient of rolling friction is mute.

01:03 What is the distance that the truck moves up the ramp before coming to a hole and solve by energy methods? so it's a good thing the problem wants us to solve by energy methods because we could solve this as if it were a chapter five problem where we draw a bunch of free body diagrams and use the kinematic equations of motion to keep track of the truck.

01:26 However, we really don't want to do that just because the math is going to get very complicated.

01:31 When we use energy methods, we only care about what's happening at two specific points.

01:38 We care about what's happening at the beginning and we care about what's happening at the endpoint.

01:43 So all we need to figure out is what is the total energy initially in the system and what is the total energy finally in the system.

01:54 So the total energy that is in the system to begin with, i'm going to bring to.

02:02 That over on the left side when the truck is starting from.

02:08 And i'm going to call that total initial energy e ei.

02:14 So that's the total energy that's going to be in the system in the first place.

02:20 And then the total energy afterwards is going to be when the truck moves up the other side of the ramp.

02:28 So i'm going to draw that at some point later.

02:31 I'm going to have the truck over here, and that's going to have gone up the ramp at some point and it's going to come to a hole.

02:39 So we want to know the distance the truck moves up that ramp.

02:44 So we have to label that, and the problem doesn't give us a variable to use.

02:49 So i'm just going to use an x to show what distance the truck moves up the ramp.

02:55 So that's the variable we're going to be solving for, is this value x, the distance.

03:01 The truck actually moves up the ramp.

03:04 And once the truck moves up the ramp, then there's going to be some total energy, some total final energy in the system.

03:14 And i'm going to denote that by ef for total final energy in the system.

03:21 So now in a perfect world where energy is perfectly and completely conserved, these two amounts of energy should be completely equal to each.

03:31 Other.

03:32 So that means that the total initial energy should equal the total final energy.

03:44 That is, of course, in a perfect world, what's happening.

03:51 Total initial energy is equal to the total final energy.

03:58 However, we're not in a perfect world in this case.

04:03 We're in a world where there exists friction.

04:06 So that means the force of friction is going to do some work on the truck as it moves up the hill, and that force of friction is going to take away some of the total initial energy in the system.

04:18 So this total initial energy, then we have to subtract whatever work is done by friction.

04:28 And then that's going to give us the total final energy that will be available.

04:34 So some energy is lost in the form of heat due to friction.

04:37 Which is what we want in bringing the truck to a halt.

04:42 So now the rest of solving this problem is just making the right substitutions into the problem.

04:53 So the first thing is we need to break down our energy terms.

04:57 So the total initial energy is composed of two things, the kinetic energy of the cart that's moving and the potential energy.

05:06 And in this case, the only source of potential energy in the system is due to gravity.

05:11 So we're going to have our gravitational potential energy.

05:14 So we know from the textbook, we know what the kinetic energy terms are.

05:20 And we know that the total initial kinetic energy, which i'm going to call ki, that's the initial kinetic energy, is one -half times the mass of the truck that we're given that is m times the speed of the truck, which were given as v .0.

05:39 So there's a basic definition for kinetic energy.

05:42 And then we know that the total initial energy, i'm going to call that ui, that is just the gravitational potential energy.

05:55 So that's going to be m times our acceleration unit of gravity, g times y and yi, what is the value for the height that the cart is above this ramp.

06:11 So we go to the truck, and i'm just going to draw the height, and i'm going to call that y -i is the total height.

06:23 Now, we can use some basic trigonometry to substitute in what we're looking at for this height, y, i.

06:31 Well, that, if we remember our trigonometry, sokatoa, the sign of alpha is equal to the alpha, opposite y i over biopotenuse which in this case is the length l that the cart has or that the truck has moved down the ramp so we can then easily write that y i is simply equal to l times the sign of alpha so we have l times the sign of alpha so we have l times the sign of alpha so that will put the potential energy term in terms of the quantities that were given, alpha and l.

07:22 So we rewrite the potential energy then, making our substitution as m times g times l times the sign of alpha.

07:38 All right, so that first step is complete.

07:42 And then we need to think about what the kinetic energy and the potential energy terms are after the truck has gone up the ramp.

07:51 Well, that final kinetic energy is just going to be equal to zero because we say that the truck has come to a halt.

08:03 We want to know how far it moves when it comes to a halt.

08:06 So its speed is going to be zero.

08:09 Therefore, its kinetic energy term must be zero.

08:12 So then we want to know what the potential energy is at this point that it's gone up.

08:20 Well, that potential energy, i'll call that uf.

08:27 Oh, excuse me.

08:34 Sorry about that.

08:36 My tablet is messing up a little bit.