Question

object 1 of mass ( m ) and object 2 of mass ( 3m ) can slide along a straight, horizontal track with negligible friction. object 1 initially moves to the right with speed ( v ) toward object 2, which is initially at rest, and the two objects collide. immediately after the collision, object 1 moves to the left with speed ( \frac{v}{2} ). is this information sufficient to determine whether the collision was elastic? why or why not?

a yes, because the fact that the velocity of object 1 changed direction during the collision means that the collision was not elastic.
b yes, because conservation of momentum can be used to determine the final speed of object 2, and the total kinetic energy of the two - object system after the collision can be calculated.
c no, because the final speed of object 2 cannot be determined without knowing whether the collision was elastic or inelastic.
d no, because the collision can be elastic for many different values of the final speed of object 2.

Explanation:

Step1: Apply conservation of momentum

The law of conservation of momentum states that \(m_1v_1 + m_2v_2=m_1v_1'+m_2v_2'\). Here, \(m_1 = m\), \(v_1 = v\), \(m_2=3m\), \(v_2 = 0\), and \(v_1'=-\frac{v}{2}\). Substituting these values into the equation:

$$
LATEXBLOCK0
$$

Step2: Calculate initial and final kinetic energies

The initial kinetic energy \(K_i=\frac{1}{2}mv^2\) (since object 2 is at rest, its initial kinetic energy is 0).
The final kinetic energy \(K_f=\frac{1}{2}m(\frac{v}{2})^2+\frac{1}{2}(3m)(\frac{v}{2})^2\)

$$
LATEXBLOCK1
$$

Since \(K_i = K_f=\frac{1}{2}mv^2\) (after using conservation of momentum to find the final velocity of object 2), we can determine if the collision is elastic (elastic collision has \(K_i = K_f\)).

Answer:

B. Yes, because conservation of momentum can be used to determine the final speed of Object 2, and the total kinetic energy of the two - object system after the collision can be calculated.