Two masses of $3.4$ kg and $2.5$ kg are accelerated from an initial speed of $5\ \text{m/s}$ and $12\ \text{m/s}$, respectively. The distances traversed by the masses in the $5^{\text{th}}$ second are $104$ m and $129$ m, respectively. The ratio of their momenta after $10$ s is $\dfrac{x}{8}$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 9
Concept. Distance in the $n$th second: $s_n=u+\dfrac a2(2n-1)$. Each mass gives its own acceleration; then $v=u+at$ gives the speed after 10 s and $p=mv$.
Step 1. Accelerations ($n=5$, so $2n-1=9$).
$$104=5+4.5a_1\Rightarrow a_1=22\ \text{m s}^{-2},\qquad 129=12+4.5a_2\Rightarrow a_2=26\ \text{m s}^{-2}$$
Step 2. Speeds at $t=10$ s.
$$v_1=5+22\times10=225\ \text{m/s},\qquad v_2=12+26\times10=272\ \text{m/s}$$
Step 3. Momenta.
$$p_1=3.4\times225=765,\qquad p_2=2.5\times272=680,\qquad\frac{p_1}{p_2}=\frac{765}{680}=1.125=\frac98$$
Answer. $x=9$.
Solution by Sreeraj P, M.Sc Physics
