# How Can Algebra Prove the Elastic Collision Equation?

In summary, this conversation is about a homework equation for elastic collisions. A teacher gave the equation (Va1-Vb1)=-(Vb2-Va1) and asks for help with solving it. A student provides a summary of the equation, which is momentum and kinetic energy are conserved. The student is still lost, and asks for help with rearranging the equations and combining them. The help is provided, and the student is now able to solve the equation.
1.

I was given the equation: (Va1-Vb1)=-(Vb2-Va1) I am suppose to prove this equation with algrebra as to why this is true. I can only use algebra, and can't use any numbers.

This is an equation for elastic collisions. There is an object;A and an Object;B. My teacher said the problem would take a page or more of algrebra to prove it. Anyone have any idea how to start the problem?

Any help is appricated.

## Homework Equations

We did a lab when we were given this problem, and the lab included the formula for kinetic energy. The lab showed in elastic collisions, momentum and Kinetic energy is conserved.

3. I honestly have no idea where to start.

What's conserved in an elastic collision?

Kinetic energy and momentum. I tried it again and I'm still lost

Kinetic energy and momentum. I tried it again and I'm still lost
That's what you need. Hint: Start by writing both equations. Rearrange each so that all the Va terms are on one side; Vb terms on the other.

Then do a bit of algebra.

So write out the Momentum equation and the Kinetic energy equation and set them equal to each other? And the final result of that should be the equation that I first posted right?

You're not going to set them equal to each other (not even sure what that means!), but you will combine them. The first step is to rearrange each equation (momentum and KE) as I suggested in the last post.

For the kinetic energy equation, there is only one m and v term. Is the m and v term in the Kinetic energy equation the v1 and m1 or v2 and m2?

Ok I set them as this-
FT+m1v1=m2v2
.5m1v2squared=.5m2v2squared

Im hoping the Kinetic Energy equation is setup right. The masses cancel out in both equations. But I can't seem to get rid of FT. Also, do I combine them and than solve to get the original equation I was given?

Your conservation of momentum equation should look something like this:

$$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$$

Your conservation of energy equation should look something like this:

$$1/2m_1v_1^2 + 1/2m_2v_2^2 = 1/2m_1v_1'^2 + 1/2m_2v_2'^2$$

That's your starting point. As a first step towards combining these equations, I recommend that you rewrite each equation, putting the terms relating to m1 on the left and the terms relating to m2 on the write.

Do that and we'll see what's next.

Ok for the initial equation, when I moved stuff around, I found that vb1=vb2. The final result I got was vb1(va1-vb1)=-vb2(vb2-va2). The vb1 and vb2 part is proved from what I first said, and what's left is the orginal equation.

Can anyone see if this makes sense?

Ok, i'll try and do what you just posted to see how that works out

Ok, I got:

ma(Va1squared-va2squared)=mb(vb2squared-vb1squared)

I got all the ma and mb terms to their respective sides.

Now do I combine that with the KE equation? Would the Ke equation be:
.5ma1va1=.5mb1vb2 ?

Since object A's intial velocity/momentum has Kinetic energy, but when it hits object b, its Kinetic energy gets transferred to object b, so object b's intial would have no Kinetic energy, but when it gets hit, it does?

Ok, I got:

ma(Va1squared-va2squared)=mb(vb2squared-vb1squared)

I got all the ma and mb terms to their respective sides.
Good. That's the rearranged KE equation. Now do the same for the momentum equation.

Ok after trying it your way, it seems I may have copied the original equation wrong.

Is it va1-vb1=-vb2-va2 as opposed to the one I first posted, where it is va1 on the right side instead of va2?

The va1 on the left and right side doesn't seem to make sense, so I think I posted the wrong intial equation.

Is there a paranthesis on the right side?

I keep getting va1-vb1=vb2-va2, I can't seem to get vb2 to be negative.

Is there a paranthesis on the right side?
The equation you want to prove, using your notation, is:
va1-vb1 = vb2-va2 = - (va2 - vb2)

Note that va1 - vb1 is the relative velocity of one mass with respect to the other. The equation you are trying to prove states that the relative velocity reverses in an elastic collision.

I keep getting va1-vb1=vb2-va2,
That's the one you want! If you've gotten that far, you're done.
I can't seem to get vb2 to be negative.
Good thing!

Thank you so much for your help! :)

## 1. What is a tough problem on elastic collisions?

A tough problem on elastic collisions is a physics concept that involves calculating the motion of objects in a collision where there is no loss of kinetic energy. This type of collision is known as an elastic collision, and it can be challenging to solve because it requires understanding of conservation of momentum and energy.

## 2. What are some real-world examples of elastic collisions?

Some real-world examples of elastic collisions include billiard balls colliding on a pool table, two cars colliding, and a golf club hitting a golf ball. In all of these scenarios, the objects involved experience a collision where there is no loss of kinetic energy.

## 3. How do you calculate the velocities of objects after an elastic collision?

To calculate the velocities of objects after an elastic collision, you can use the equations of conservation of momentum and conservation of kinetic energy. These equations involve the masses and velocities of the objects before and after the collision, as well as the coefficient of restitution, which represents the elasticity of the collision.

## 4. What is the difference between an elastic collision and an inelastic collision?

The main difference between an elastic collision and an inelastic collision is that in an elastic collision, kinetic energy is conserved, whereas in an inelastic collision, kinetic energy is not conserved. In an inelastic collision, some of the kinetic energy is converted into other forms of energy, such as heat or sound.

## 5. Why is it important to understand elastic collisions?

Understanding elastic collisions is important because it allows scientists and engineers to predict and analyze the motion of objects in various scenarios, such as in sports, transportation, and industrial processes. It also helps in the development of safety measures and efficient designs to minimize the impact of collisions.

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