A 5.00 g bullet is fired horizontally into a 1.20 kg wooden block resting on a horizontal surf

step 1 Welcome to a new problem. This time we're given a block that's on a sliding surface. So it's sit

A 5.00 g bullet is fired horizontally into a 1.20 kg...

Key Concepts

Kinetic Friction

Kinetic friction is the resistive force acting opposite to the direction of motion when an object slides over a surface. It is computed as the product of the coefficient of kinetic friction and the normal force, and it is crucial in analyzing how energy is dissipated during the motion of a sliding object.

Work-Energy Principle

The work-energy principle relates the work done on an object to its change in kinetic energy. In dynamics problems involving friction, the work done by the frictional force (calculated as the product of the friction coefficient, normal force, and distance traveled) equates to the loss of kinetic energy, determining how far the object travels before stopping.

Conservation of Momentum

This principle states that, in the absence of significant external forces during a collision, the total linear momentum of the system remains constant. In situations like a bullet embedding into a block (a perfectly inelastic collision), momentum before impact is equal to the combined momentum immediately after impact, even though kinetic energy is not conserved.

Perfectly Inelastic Collisions

These collisions are characterized by the colliding objects sticking together after impact, which maximizes energy loss through deformation or heat. While momentum is conserved, the mechanical kinetic energy is reduced, and additional analysis (such as energy dissipated by friction) is needed to understand the motion post-collision.

Transcript

00:01 Welcome to a new problem.

00:03 This time we're given a block that's on a sliding surface.

00:11 So it's sitting at rest on a sliding surface.

00:15 That means that the initial velocity of the block is zero meters per second.

00:23 So it's not moving.

00:25 That's what we're given.

00:26 And then also we're told that the coefficient of kinetic friction for this surface is 0 .2.

00:36 Another piece of information that we're given is the mass of the block, which happens to be 1 .20 kilograms.

00:49 We're also given a bullet.

00:53 So there's a bullet that's fired.

00:56 So this is kind of like the before, like, you know, what happens before.

01:00 So a bullet, the mass of the bullet, we're going to call it l, is given us 5 grams.

01:12 And we can change that 5 grams right away to kilograms by saying this is multiplied by one kilogram of a thousand, a thousand grams, and that gives us 5 times 10 to the negative 3 kilograms.

01:30 That's the mass of the bullet.

01:31 The bullet is moving with an initial velocity that happens to be the, so while the bullet has an initial velocity, we still don't know what that is, and that's what we want to find out.

01:53 You know, what's the initial speed of the bullet? that's our goal.

01:58 Given all this information.

02:01 The second aspect of the problem is once the bullet hits the block and sits inside of it, the block starts to move in the right direction.

02:18 And so the outcome of this movement is a distance travel equivalent to, so we're going to call that s equals to 0 .310 meters.

02:37 That's the distance traveled.

02:39 It's going to come to a stop.

02:40 So the block is going to stop at some point.

02:44 Okay, the block is going to stop at some point.

02:49 And that's the distance covered.

02:53 There's almost some, i would say some resistance from friction.

02:59 So there's a frictional force pointing in the opposite direction of the motion, f of f.

03:06 And we can always compute that friction.

03:09 Okay, we can compute that frictional force.

03:13 So two ideas we're going to use.

03:15 We're going to use the law of conservation of momentum.

03:22 And also we're going to use the computations for frictional force.

03:27 So that's the first thing we want to compute.

03:29 You know, what's the friction of force between the block and the surface? remember, the force of friction happens to be, you know, we need the frictional force because it's going to help us determine the work done by friction.

03:49 So we need that.

03:51 We have the frictional force is mu times the normal force.

03:56 Remember, when this thing is moving, there's a normal force pointing that way, and then there's a weight, you know, the weight of the block.

04:07 Now here it's going to be the weight of the block and the bullet.

04:14 So we have the mass of the block plus the mass of the bullet together.

04:21 So, you know, the first thing we want to do is obviously compute that frictional force.

04:26 So this becomes f of f equals to mu times the mass.

04:33 You know, this is kind of like the combined mass of the block.

04:38 And the bullet times the acceleration due to gravity.

04:45 So that's the information we're given.

04:49 And then on top of that, we want to find the work done by friction, and that's just the frictional force times the displacement, this s displacement right here.

05:03 And so the work done by friction is mu times mb plus ml, times s.

05:14 That's the work done by friction.

05:17 The second part of the problem, we have to, we have, so our goal is to find the initial velocity of the bullet.

05:29 Law of conservation of energy.

05:32 So here we're using law of conservation of energy.

05:37 And we're saying that the, so we're using two laws.

05:43 You know, first of all, law of conservation of energy.

05:45 Momentum, we're using law of conservation of momentum, and then also, you know, the work energy theorem.

05:56 So let's kind of like clarify that.

06:00 We want to know what's, so the, so there are two components here.

06:05 So we have, we have the work done by friction, done by friction.

06:16 It's going to be equivalent to the change in kinetic energy.

06:23 Of the system which is the work done by the bullet and the block so it's kind of like that we already have work done by friction mu so so here we'll write work done by friction is equal to work done by the bullet plus the block and work done by friction from this side is mu mbml, or mb plus ml, so mass of bullet and mass of block, g .s.

07:12 And then we want to find the change in kinetic energy of the system...

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