Question

Let two objects of equal mass m collide. Object 1 has initial velocity v, directed to the right, and object 2 is initially stationary.
A. If the collision is perfectly elastic, what are the final velocities v1 and v2 of objects 1 and 2?

Give the velocity v1 of object 1 followed by the velocity v2 of object 2, separated by a comma. Express each velocity in terms of v.

Answer 1

Answer:

(v, 0) or (0, v)

Explanation:

The law of conservation of linear momentum states total initial momentum equal total final momentum.

Total initial momentum = $mv + m\times 0 = mv$

Total final momentum = $mv_1 + mv_2 = m(v_1+v_2)$

Equating both,

$mv = m(v_1+v_2)$

$v = v_1+v_2$

For an elastic collision, kinetic energy is conserved i.e. total initial kinetic energy = total final kinetic energy

$\frac{1}{2}mv^2 + \frac{1}{2}m\times0^2 = \frac{1}{2}mv_1^2 + \frac{1}{2}mv_2^2$

$v^2 = v_1^2+v_2^2$

From the first equation, $v_1 = v-v_2$.

Substituting this in the second equation,

$v^2 = (v-v_2)^2+v_2^2$

$v^2 = v^2-2vv_2 +v_2^2+v_2^2$

$0 = 2v_2^2-2vv_2$

$v_2(v_2-v) = 0$

$v_2 = 0$ or $v_2 = v$

From $v_1 = v-v_2$,

$v_1 = v-v = 0$

OR

$v_1 = v-0 = v$