A point mass M moving with a certain velocity | vedclass

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(D) Since the collision is elastic,both linear momentum and kinetic energy are conserved.Let $u_1$ be the initial velocity of mass $M$,and $v_1, v_2$

Vedclass · Accepted Answer

$1/4$

Vedclass · Answer

(D) Since the collision is elastic,both linear momentum and kinetic energy are conserved.Let $u_1$ be the initial velocity of mass $M$,and $v_1, v_2$ be the final velocities of masses $M$ and $M/2$ respectively.Conservation of linear momentum:$M u_1 = M v_1 + (M/2) v_2$$u_1 = v_1 + v_2/2$$2 u_1 = 2 v_1 + v_2$ ... $(i)$Conservation of kinetic energy:$(1/2) M u_1^2 = (1/2) M v_1^2 + (1/2) (M/2) v_2^2$$u_1^2 = v_1^2 + v_2^2/2$$2 u_1^2 = 2 v_1^2 + v_2^2$ ... (ii)From $(i)$,$u_1 = (2 v_1 + v_2) / 2$. Substituting this into (ii):$2 ((2 v_1 + v_2) / 2)^2 = 2 v_1^2 + v_2^2$$2 (4 v_1^2 + v_2^2 + 4 v_1 v_2) / 4 = 2 v_1^2 + v_2^2$$(4 v_1^2 + v_2^2 + 4 v_1 v_2) / 2 = 2 v_1^2 + v_2^2$$4 v_1^2 + v_2^2 + 4 v_1 v_2 = 4 v_1^2 + 2 v_2^2$$4 v_1 v_2 = v_2^2$Since $v_2 
eq 0$,we have $v_2 = 4 v_1$.The ratio of final velocities of $M$ and $M/2$ is $x = v_1 / v_2 = 1 / 4$.

$A$ point mass $M$ moving with a certain velocity collides with a stationary point mass $M / 2$. The collision is elastic and in one-dimension. Let the ratio of the final velocities of $M$ and $M / 2$ be $x$. The value of $x$ is

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