Questions 1, 2, and 3 are short free-response questions that...

Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part. (DIAGRAM IS NOT AVAILABLE TO COPY) A mass $m_{1}$ traveling with an initial velocity of $v$ has an elastic collision with a mass $m_{2}$ initially at rest. (a) Determine the final velocity $v_{1}$ of $m_{1}$ in terms of $m_{1}, m_{2},$ and $v$ (b) Determine the final velocity $v_{2}$ of $m_{2}$ in terms of $m_{1}, m_{2},$ and $v$ (c) For what values of $m_{1}$ and $m_{2}$ would the final velocities of the two masses be in the same direction? The opposite direction? (DIAGRAM IS NOT AVAILABLE TO COPY)

Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.
(DIAGRAM IS NOT AVAILABLE TO COPY)
A mass $m_{1}$ traveling with an initial velocity of $v$ has an elastic collision with a mass $m_{2}$ initially at rest.
(a) Determine the final velocity $v_{1}$ of $m_{1}$ in terms of $m_{1}, m_{2},$ and $v$
(b) Determine the final velocity $v_{2}$ of $m_{2}$ in terms of $m_{1}, m_{2},$ and $v$
(c) For what values of $m_{1}$ and $m_{2}$ would the final velocities of the two masses be in the same direction? The opposite direction?
(DIAGRAM IS NOT AVAILABLE TO COPY)

Key Concepts

Elastic Collisions

Elastic collisions are interactions between objects in which both total momentum and total kinetic energy are conserved. This distinguishes them from inelastic collisions, where some kinetic energy is lost. Understanding elastic collisions requires applying conservation principles to derive relationships between the initial and final velocities of the objects involved.

Conservation of Momentum

The conservation of momentum states that the total momentum of a closed system remains constant over time, provided no external forces act on it. In collision problems, this principle is used to set up an equation that links the momenta of the objects before and after the collision, providing a necessary condition for solving for unknown velocities.

Conservation of Kinetic Energy

Conservation of kinetic energy indicates that in an elastic collision, the total kinetic energy before and after the collision remains the same. This principle, combined with momentum conservation, yields a second independent equation allowing for the unique determination of the final velocities of colliding bodies.

Determining Motion Direction

Determining whether the final velocities of the objects are in the same or opposite directions involves analyzing the signs of the final velocity expressions derived from the conservation equations. This concept is fundamental to understanding the outcome of collisions, as the relative magnitudes of masses and the resulting velocity signs indicate the direction of motion post-collision.

Transcript

00:01 In this example, we're going to analyze what's called a perfectly elastic collision.

00:07 And what we mean by that is both total momentum is conserved.

00:15 So the initial equals the final and kinetic energy is conserved.

00:25 We'll start with a simple example where we have a particle m1 moving in one dimension with velocity v.

00:34 And it will collide with a second particle mass two.

00:39 That is initially at rest.

00:42 So that's our initial situation.

00:45 To make life easy, i'm going to draw my final situation as both particles are now moving off together to the right with new velocities.

01:05 So how do we analyze this? we have two equations that we can work with.

01:10 The momentum conservation will look like m1 times v equals m2 v2 plus m1 v1.

01:24 And ordinarily momentum is a vector.

01:28 We do not have to worry about that because we're in one dimension, moving along the x direction.

01:36 However, if one of the velocities were going to the left, we'd have to have to.

01:42 To include a negative sign.

01:46 And our solution could provide such an answer.

01:50 Conservation of kinetic energy 1 half m1 v squared is equal to the sum of the kinetic energies afterwards.

02:08 So two equations.

02:11 What do we do with those? whoops, that should be a 1.

02:14 What do we do with those? what we are going to do with those? is we are going to get an expression for the v1 and the v2 in terms of our initial velocity.

02:30 So we're going to treat v as a known, and we're going to treat the two after -effect velocities as our unknowns.

02:42 And use these two equations to develop a relationship for both of those.

02:48 And the algebra can get a little tricky, mostly because we have that quadratic nature of the kinetic energy.

02:59 So i will show you the algebra trick is to bring over on the kinetic energy equation the two terms with the m1 in them.

03:13 And what that buys you is it gives you this difference between two squares, which we can expand out as 1⁄2m1 times the difference between the two velocities and their summation.

03:37 Now, that may not look like it gets you anything, but we can do a similar thing with equation 1.

03:44 We can bring over the v1 term, and we can then, i'll show you what the trick is, we're going to, solve the one equation, the momentum equation, for v minus v1.

04:10 And here's the really nice part.

04:13 We can then substitute this back into our kinetic energy.

04:22 And believe it or not, that is going to make our life very nice.

04:30 It's hard to believe, but it definitely will make our life nice.

04:40 So let's see how that works.

04:51 Okay, so we'll put those back into equation number two and kind of clean up equation number two...

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