**Example:** Elastic Collisions

Hey guys, so let's check out this example of elastic collisions, now remember elastic collisions are special in that 1) kinetic energy is conserved so K final=K initial and 2) we can use an extra equation which is V1 initial+V1 final=V2 initial+V2 final, I want to talk about the situation real quick we're not actually going to use this is conceptual, this is what we're actually going to use as an extra equation and by the way the second question comes from the first equation this is just a useful version of the first one, OK? I want to point out something real quick this is different from this equation because first of all there are no masses but the other difference is that here you have initial 1, initial 2, final 1, final 2, here you have initial final, initial final So here it's 1,2 1,2 here's 11,22 so it's a little but different you got to figure out a way to remember that, so 1, 2, 1, 2 and 1, 1 2, 2, OK? So let's do this, 2 objects of the same mass undergone an elastic head on collision, the reason I'm able to use the second equation in addition to the first equation is because it told me here that this is elastic, head on just means that they're going towards each other like this, they have the same mass it doesn't tell me what the mass is so I'm just going to call it M for now, object A from the left (so this has to be A and therefore this is B) has speed of 3 and object B has speed of 4, remember opposite directions opposite signs +3-4 please don't forget the negative 4 or you're going to get the wrong answer, calculator final velocities, magnitude and direction, magnitude means that number and direction means positive or negative, OK? How are we going to do this? it's a collision conservation of momentum so M1 V1 or in this case Ma Va+Mb Vb both initial=Ma.Va+MbVb final, I know the masses......Actually I don't know the masses but they're all M so what's going to happen is whenever the 2 masses are the same you're going to be able to cancel it, right? So, whether it's M and M or if it's 10 and 10 whatever if the numbers are the same you can always cancel it from this equation, Va initial is right there=+3, Vb initial is -4. Va final I don't have and we're looking for it and Vb final I don't have and we're looking for it, so what we ask here could be well they just didn't give me enough information? But they did instead of giving one of these two velocities they replaced that with the information that this is elastic so it's a give and take, I tell you the type of collision but then I give you one less number for you to be able to solve this problem, OK? So instead of giving you a number I told you that it was elastic and you can still solve it but this is 2 unknowns with just one equation which means you can't solve it, what do you do? Well you need another equation so the other question you are going to use is this one right here, OK? So, you're going to use the equation, the special equation for elastic collision so the V1 initial + V1 final= V2 initial + V2 final, by the way there are a few different versions of this equation some of them have like negatives in them what not, I moved things and I showed it to you this way because I think it's the easiest one to remember but you might see it slightly different, alright? So let's plug in the numbers, the initial velocity here...The initial velocity of 1=+3, the final velocity of 1 I don't have it and this one is a -4 and this one I don't have it, so now that I have two equations I can actually solve this by combining the two equations so the first thing I'm going to do is I'm going to combine the numbers and put the variables in the same sides, so 3-4=-1 Va final+Vb final, there's nothing else I can do here for now, let's do the same thing here let's move this number over here so that I have numbers on the left and letters in the right, so the 4 goes over here I got a 7 equals and then this guy goes over here and then it's going to a negative, negative this is A and B actually, right? -Vaf +Vbf, now notice that these two equations have the same basic structure, number an A and the B, a number the A and the B, OK? So, what I can do now is I can move this guy over here and we're going to combine these two equations and the easiest way to do this is by adding the equations so let's stack them up on top of each other. You can add equations remember that and the way that you add equations is everything to the left just as a quick reminder everything to the left of the equals sign gets added so -1+7=+6plus and everything to the right of the equals sign gets added so I have Va+-Va so Va+-Va just cancels equals Vb+Vb that's 2 Vb, 2Vbf which means Vbf is 6/2=3, +3, Vbf=+3. Now that I got one of the two velocities I can simply get that number and plug it into either one of the 2 original equations, Ok? To find Vaf, I'm going to plug into this one here just because that equation the Va is already positive, OK? Let's move this over here and I have a -1 equals... Whoops Vaf+Vbf, -1 Vbf is 3, it's going to go to the left as -3, Vaf=-4 meters per second, OK? This is positive so it's going to the right so B after the collision is going to go to the right with 3 and A after the collision is going to go to the left with 4, OK? You might have noticed that the numbers flipped, this guy was a 3 ended up with a 4, this guy is a 4 and ended up with a 3 that's what happens if you have elastic collisions but only happens if you have exactly the same mass, the velocity simply flip, OK? So that's it for this one, elastic collisions are a little bit more complicated a little bit more work because you do have fewer numbers that are going to be given to you and you do have to use two equations to solve them, there is the big equation, the big momentum equation and you got to use the extra elastic collision equation, so that's it let me know if you have any questions.

An 54.5 kg object moving to the right at 55.9 cm/s overtakes and collides elastically with a second 35.7 kg object moving in the same direction at 39 cm/s. Find the velocity of the second object after the collision.

1. 39.2634

2. 56.5027

3. 38.8214

4. 55.3243

5. 59.4224

6. 39.9813

7. 58.3122

8. 57.2106

9. 41.2414

10. 48.0473

Watch Solution

In the “slingshot effect,” the transfer of energy in an elastic collision is used to boost the energy of a space probe so that it can escape from the solar system. All speeds are relative to an inertial frame in which the center of the sun remains at rest. A space probe moves at 6.4 km/s toward Saturn, which is moving at 6.4 km/s toward the probe. Because of the gravitational attraction between Saturn and the probe, the probe swings around Saturn and heads back (nearly) in the opposite direction with speed *v _{f}*. By what factor is the kinetic energy of the probe increased?

1. 16.2591

2. 22.1363

3. 9.14008

4. 4.80263

5. 6.404

6. 7.94215

7. 3.7032

8. 9.0

9. 3.61

10. 9.33331

Watch Solution

A projectile of mass m _{1} moving with a speed v_{1} in the +x direction strikes a stationary target of mass m_{2} = 2m_{1} head-on in an elastic collision. Find the final velocity of the projectile m_{1}.

Hint: You can use the energy and momentum principles.

1. -1/3 v_{1}

2. -1/4 v_{1}

3. -1/5 v_{1}

4. 1/3 v_{1}

5. -1/2 v_{1}

6. v_{1}

7. -5 v_{1}

8. 3 v_{1}

Watch Solution

A 5 kg and a 3 kg mass collide head on, with the 5 kg mass moving at an initial speed of 15 m/s and the 3 kg mass moving at an initial speed of 10 m/s. Answer the following questions:

a) If the collision is completely inelastic, what is the final speed and direction of each mass?

b) If the collision is elastic, what is the final speed and direction of each mass?

Watch Solution