How Does Mass Affect Distance in a Two-Cart Air Track Experiment?

[gregcor]

Homework Statement

Two carts are set up on an air track with a rubber band connecting them. The carts are pulled just slightly more than the length of the relaxed rubber band. Upon release, mass 1 travels a distance of x₁ and mass 2 travels a distance of x₂. Show that $$\frac{x_1^2}{x_2^2}=\frac{m_2}{m_1}$$

Homework Equations

$$F_s=-kx$$

$$x=x_1+x_2$$

$$p=mv$$

The Attempt at a Solution

I'm not exactly sure where the first two equations come in. However, I noticed:
$$m_1 v = m_2 v$$

$$m_1 \frac{x_1}{t} = m_2 \frac{x_2}{t}$$

$$\frac{m_1^2}{x_2^2} = \frac{m_2^2}{x_1^2}$$


Though that's clearly not the original equation. I'm a bit lost at where I should go from here.

Thanks!

 

[anathema]

Thank you for your question. It seems like you are on the right track with your attempt at a solution. The first two equations are used to describe the forces acting on the carts and the relationship between their positions.

To further progress with the problem, we can use the fact that the rubber band connecting the carts exerts a force on both of them, causing them to accelerate in opposite directions. This force can be described by Hooke's law, F_s = -kx, where F_s is the force of the rubber band, k is the spring constant, and x is the displacement from its relaxed length.

Using this equation, we can set up the following equations:

F_s = m_1a_1

F_s = m_2a_2

where a_1 and a_2 are the accelerations of mass 1 and mass 2, respectively.

We can also use the fact that the total displacement of the two carts is equal to the relaxed length of the rubber band, x = x_1 + x_2.

Now, we can substitute the equations for F_s into the equations for acceleration and solve for x_1 and x_2.

m_1a_1 = -kx_1

m_2a_2 = -kx_2

a_1 = \frac{-kx_1}{m_1}

a_2 = \frac{-kx_2}{m_2}

Substituting these expressions for acceleration into the equation for x = x_1 + x_2, we get:

\frac{-kx_1}{m_1} + \frac{-kx_2}{m_2} = x

Solving for x_1 and x_2, we get:

x_1 = \frac{m_2}{m_1+m_2}x

x_2 = \frac{m_1}{m_1+m_2}x

Now, we can substitute these values for x_1 and x_2 into the original equation, \frac{x_1^2}{x_2^2} = \frac{m_2}{m_1}, and we get:

\frac{(\frac{m_2}{m_1+m_2}x)^2}{(\frac{m_1}{m_1+m_2}x)^2} = \

 

[nlightner]

I would approach this problem by first looking at the physics principles involved. The first equation, F_s=-kx, is known as Hooke's law and describes the force exerted by a spring (or in this case, a rubber band) as a function of its displacement. The second equation, x=x_1+x_2, simply states that the total distance traveled by the two carts is equal to the sum of their individual distances.

To solve this problem, we can use the conservation of momentum principle, which states that the total momentum of a system remains constant unless acted upon by an external force. In this case, the only external force is the tension in the rubber band, which is equal for both carts.

Using the equation p=mv, we can rewrite the given equation m_1v=m_2v as m_1(x_1/t)=m_2(x_2/t). This is because the velocity of each cart is equal to its distance traveled divided by the time it takes to travel that distance.

Now, we can rearrange this equation to get x_1/x_2 = m_2/m_1. This means that the ratio of the distances traveled by the two carts is equal to the ratio of their masses.

Finally, we can square both sides of this equation to get x_1^2/x_2^2 = m_2^2/m_1^2. And since we know that x=x_1+x_2, we can substitute this into the equation to get x_1^2/x_2^2 = (x-x_1)^2/x_2^2.

Simplifying this further, we get x_1^2/x_2^2 = (x^2-2xx_1+x_1^2)/x_2^2. Canceling out the x_1^2 terms and rearranging, we get x_1^2/x_2^2 = (x^2/x_2^2) - (2x_1/x_2).

Since we know that x_1/x_2 = m_2/m_1, we can substitute this into the equation to get x_1^2/x_2^2 = (x^2/x_2^2) - (2m_2/m_1).

Finally, rearranging this equation gives us the desired result