2. Two objects of equal mass collide in a glancing collision...

2. Two objects of equal mass collide in a glancing collision resulting in deformations to both objects. Before the collision, the first object (let's call it 'A') was seen traveling at 35-m/s; the other object (let's call it 'B') was stationary. After the collision object A is deflected by an angle of 30° relative to its original path of travel; object B on the other hand travels at an angle of 40° relative to the original path of object A. a. Determine the speed of each object after the collision (15pts) b. Determine how much energy the system had before and after the collision. How much energy (if any) was lost in the collision process? Express the energy loss as a percentage. (10pts)

Transcript

00:01 So this problem gives us a glancing collision between two objects of equal mass.

00:06 Object a is moving at 35 meters per second and glances into object b.

00:12 Object a moves away upward at a 30 degree angle with some speed that i'm going to call va.

00:18 And object b is going to be pushed downward and to the right at a 40 degree angle with some speed that we will call vb.

00:28 Our goal is to find how fast both of these objects move away, and then find the initial and final energies of the system and how much energy was lost as a percentage.

00:40 So anytime we have a glancing collision like this, we're dealing with a two -dimensional conservation of momentum problem.

00:47 So since object a is moving just in the x direction, i'm only going to consider conservation of momentum for the x direction.

00:56 And eventually we will figure out what happens in the y direction.

01:02 But for now, this just tells us that the total momentum of the system in the x direction is going to equal the final momentum of the system in the x direction, or momentum is conserved in the x direction.

01:14 So if we look at the total initial momentum, only object a is moving.

01:20 So object a is going to be responsible for the total momentum before the collision.

01:26 It tells us the mass of both objects is m.

01:30 So the momentum mass times velocity is just going to be m times 35.

01:36 Object b has no initial momentum.

01:39 And then after the collision, object a has this component of velocity in the x direction, or it has this component of momentum in the x direction.

01:53 So we can say for after the collision, object a has a momentum of mass times its velocity in the x direction, which is just va cosine theta.

02:09 And then we can do the same thing for object b, since object b has this component to velocity in the x direction, it will also have a component of momentum in the x direction, mass times its x component of velocity, or vb, cosine, it.

02:28 So we can clean this up a little bit, so we'll say 35m equals m times va cosine of 30, plus m for the momentum for b times vb cosine of 40.

02:52 And we see that there's a mass in each term.

02:55 So if we just divide both sides by m, the masses reduce out.

02:59 So we get 35 equals va, cosine 30 plus vb cosine 40.

03:10 So as it stands now, we have one equation with these two unknowns.

03:14 We can't solve this just yet.

03:16 So we'll need another equation that has va and vb in it.

03:21 So this is where the conservation of momentum in the y direction will come into play.

03:26 So if we look at the initial momentum and the final momentum in the vertical direction or in the y direction, hopefully we'll be able to end up with another equation that has va and vb in it, and then we can solve the system of equations.

03:42 So initially, none of these objects are moving vertically, so there is no momentum initially in the y direction, which means after they collide, this vertical component of motion for object a has to cancel out the momentum or the vertical momentum of vb in the negative.

04:02 Y direction.

04:04 So we can get an equation that says the mass for a times its vertical component of velocity, va, sine theta, minus, and i'm using minus instead of addition because this vector points downward, so this will have a negative component for momentum.

04:27 So minus the mass times this vertical component, which is vb, sine theta.

04:35 So both of these vertical components of momentum have to be equal to each other in order to cancel each other out and result with no momentum.

04:47 So momentum still has to be conserved.

04:49 If we start with no vertical momentum, we have to end with no vertical momentum.

04:53 So we can see that these two masses will cancel out again, and we can find va sine of 30 equals vb sign of 30.

05:11 Oh, sorry, sign of 40.

05:17 So now we have two equations with the same two unknowns, vb, va, vb, va, vb, va, and we can solve this for one of these variables and then plug it into the other equation...

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