You're investigating an accident in which a 1040 -kg Toyota Yaris and an 2140 -kg Buick Enclave

You're investigating an accident in which a 1040 -kg Toyota...

Key Concepts

Work-Energy Principle

This principle states that the work done by forces on an object is equal to the change in its kinetic energy. In accident reconstruction, the frictional work performed by the road (through frictional forces acting over the skid distance) is set equal to the initial kinetic energy of the combined wreckage. This relationship allows investigators to estimate the speed before the vehicles came to a stop.

Kinetic Friction

Kinetic friction is the force that opposes the motion of objects sliding over a surface. It is computed as the product of the coefficient of friction and the normal force (which is proportional to the object's weight). In the context of vehicle accidents, the kinetic friction between the tires (or wreckage) and the road dissipates the vehicle’s kinetic energy during skidding, thereby providing a measurable link between skidding distance and initial speed.

Deceleration and Stopping Distance

When an object is subject to friction, it experiences a deceleration that is proportional to the coefficient of friction and the acceleration due to gravity. By applying the equation v² = 2ad (where a is the deceleration and d is the stopping distance), one can compute the initial velocity required to produce the observed skid distance. This calculation is critical in determining if a vehicle was traveling above a certain speed limit at the time of the accident.

Collision Dynamics and Inelastic Collisions

In real-world vehicle accidents, collisions are often inelastic, meaning that the vehicles do not rebound off each other but instead stick together and move as a single unit immediately after impact. The combined mass and resulting kinetic energy of the wreckage are then used in energy dissipation calculations. Investigators analyze this post-collision energy to backtrack and estimate the pre-collision speeds of the vehicles involved.

Transcript

00:01 So in this problem, we're dealing with a collision, and we're trying to find the show that one car exceeded the speed limit.

00:12 So i'm going to go ahead and get the givens down, and then i'll talk about an approach to the problem.

00:17 So mass of yaris, mass of 1040, mass of buick is 2140, and they're going up.

00:46 At right angles and they skid 12 .3 meters, so x is 12 .3 meters.

00:57 And the coefficient is 0 .712.

01:07 You want to show at least one had a bigger speed than 55 kilometers per hour.

01:12 It's kind of interesting.

01:14 So it's kind of draw the diagram.

01:18 So one's coming in here, one's coming in there, and then they have to.

01:23 Skid, i guess, at, you can see it's going to be up into the right, so they're going to skid until they stop.

01:31 So let's just sort of apply the laws and see what we can calculate from that, what equations we can get.

01:38 So we can do momentum conservation because at the time of the collision, we're assuming that the collision happens instantaneously.

01:45 There's no extra forces being applied, so there's no changes in momentum of the system.

01:50 So we can say momentum is conserved.

01:52 So we can say p initial is p final.

01:59 So let's say that p initial is going to be the mass of the yaris times the velocity of the yaris.

02:09 And then let's say that the, let's say this is the yaris and then this is the buick.

02:15 Oh, actually, to make it less confusing, i'm going to say that the yaris is going vertically so i can be going in the y direction.

02:24 And the buick is going horizontally.

02:28 So the yaris is going in the j -hat direction then, and then the buick is going to go in the i -hat direction, and that's equal to their final momentum, which is mass of buick plus mass of yaris, times their final velocity, which will be up into the right.

02:59 So this equation's true.

03:00 It doesn't really get us anything so far.

03:05 It just gives us a y and a b, a vy and vb.

03:08 We don't know v either.

03:10 So there's three in notes.

03:11 So it's not really going to help us right now.

03:12 But what i think we can do is let's focus on the skitting.

03:20 And so we'll put this momentum equation aside for later because i'm sure we'll use it.

03:24 And now we'll focus on skitting.

03:26 So while it's skitting, we know that it's taking all the kinetic energy that it has and turning it into internal energy by the time it stops.

03:35 And so we can say that the initial kinetic energy, so one -half, buick plus mass of the yaris, let's call this the final, the final squared, final just because it's after the collision, although it's not the final thing that happens in this problem.

04:00 And that's going to be equal to the work done by friction.

04:03 So it's going to be mu times the normal force, times the distance.

04:07 So the normal force is just going to be the mass times gravity.

04:14 So mb plus m y gravity.

04:19 So there's mu times the normal and then times the distance.

04:23 Now look at this.

04:27 These cancel.

04:30 So what we can, this equation is, more immediately useful because we can get the final velocity, right? we have mu, we have x, we have g, we have one half.

04:41 So we can say the magnitude of v is equal to 2 mu g x square root.

04:56 So if i wanted, i can stop and get the number.

04:59 I'm just going to kind of take a second to pause and think how to go about this.

05:04 So it looks like from here, i can take this and plug this in here.

05:09 And i don't know the exact direction yet, but i can actually say the magnitude of the final momentum.

05:18 And so then what i can do is then once i have the final momentum, i can say it's equal to the initial momentum.

05:29 And so i can show that at least one of these sides had to be.

05:36 55 kilometers per hour.

05:39 Sorry, i'm not sure that totally made sense, but i'm going to proceed anyway, and then maybe it'll become more clear what exactly it means.

05:51 And now i'm just going to pause the debate if it would be useful to go ahead and do numbers.

05:56 I think it would be useful to do some numbers.

05:59 Usually, sometimes i like to just do equations, but i think with all the logic statements in there, it's going to get a little messy.

06:07 So let's just go ahead and put numbers to this.

06:10 So we want to get the square root of 2 times 9 .81 times 0 .712.

06:19 Let's go to double check my numbers times .712 times x which is 12 .3.

06:35 So 1, 2, 3, yes.

06:37 And so i get that it's 13 .1 meters per second.

06:53 And then i can go ahead and plug this in here and multiply by the mass of the yaris plus the buick.

06:59 So 1040 plus 2140...

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