A car of mass m moving at a speed v1 collides and couples...

A car of mass $m$ moving at a speed $v_{1}$ collides and couples with the back of a truck of mass 2 $\mathrm{m}$ moving initially in the same direction as the car at a lower speed $v_{2} .$ (a) What is the speed $v_{f}$ of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car-truck system in the collision?

Key Concepts

Kinetic Energy Loss in Collisions

In inelastic collisions, the system's kinetic energy decreases because some of it is dissipated due to deformation, friction, and other non-conservative forces. The change in kinetic energy is determined by comparing the total kinetic energy before the collision with that after the collision, highlighting the efficiency—or lack thereof—in energy transfer during impact.

Inelastic Collision

An inelastic collision is a type of collision in which the colliding objects stick together after impact. While momentum is conserved, kinetic energy is not; some is transformed into other forms such as heat or sound. This concept is essential when calculating the final velocity of coupled bodies and determining the associated kinetic energy loss.

Conservation of Linear Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. In collision problems, like when two vehicles lock together, conservation of momentum is applied to relate the combined mass and velocity after the collision to the individual momenta before impact.

Transcript

00:01 Okay, so here you're given a car that's moving in an direction with speed v1 and mass m.

00:11 And then you have another car moving in the same direction with speed v2 in mass 2m.

00:19 And you're told that they collide and then finally they each move with the same speed because you're now stuck together v -final.

00:30 And you have to calculate this final velocity and you have to calculate the change in kinetic energy so in this case we're going to use momentum equations first because you know pi again is equal to pf regardless of what collision it is this is going to be an inelastic collision because these things stick together afterwards so the total mass here will be 3m because it's 2m plus m since they got stack together so for pi it's just going to be m v1 plus 2m v2 should be equal to 3m v final and if you simplify this you realize that v final is actually equal to again this m is common to both expressions so i can just cancel that and then this should be v1 plus v2 over three so that's your final velocity there unless you're given any one of these two you cannot solve for the value of vf and in this case you have three unknown variables here so you will not be able to calculate the v -final numerically because you need three different equations to solve for this.

02:12 So now the next part, this thing wants to calculate changing the kinetic energy, but that's just the kinetic energy that you had initially minus the kinetic energy that you're going to have at the end.

02:28 And this equation essentially becomes again the kinetic energy initially was just half m v1 squared plus half 2m again this is 2 v2 squared or the 2 here because it's 2m minus this final kinetic energy here which is half of 3m v final squared and notice that now you just you don't this is just calculating the change in kinetic energy here so now you go on to simplify this so we know that so here you can go out to simplify this so we know that so here you can go out and remove that to there and we know what this is in terms of v1 and v2.

03:50 So if you want, you can just go ahead and plug that in to simplify it further.

03:58 So that's now mv2 squared minus 3 over 2m...

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