Question

Two masses of 3.4 kg and 2.5 kg are accelerated from an initial speed of 5 m/s and 12 m/s, respectively. The distances traversed by the masses in the 5th second are 104 m and 129 m, respectively. The ratio of their momenta after 10 s is \(\frac{x}{8}\). The value of \(x\) is ________.

Answer 1

Correct Answer: 9

Step 1: Understanding the Concept:
We are given distance traveled during a specific individual second (the 5th second). By using the formula for distance traveled in the $n^{th}$ second, we can calculate the constant acceleration of each mass. Then, using kinematics, we find their final velocities at $t=10\text{ s}$ to calculate the ratio of their final momenta.
Step 2: Key Formula or Approach:
1. Distance traveled in the $n^{th}$ second: $S_n = u + \frac{1}{2}a(2n - 1)$.
2. Velocity after time $t$: $v = u + at$.
3. Momentum: $p = m \times v$.
Step 3: Detailed Explanation:
For Mass 1:
Mass $m_1 = 3.4\text{ kg}$, initial speed $u_1 = 5\text{ m/s}$.
Distance in 5th second $S_{5,1} = 104\text{ m}$.
$104 = 5 + \frac{1}{2} a_1 (2 \times 5 - 1) = 5 + \frac{9}{2} a_1$.
$104 - 5 = 4.5 a_1 \implies 99 = 4.5 a_1 \implies a_1 = \frac{99}{4.5} = 22\text{ m/s}^2$.
Velocity of mass 1 after 10 seconds:
$v_1 = u_1 + a_1(10) = 5 + 22(10) = 5 + 220 = 225\text{ m/s}$.
Momentum of mass 1: $p_1 = m_1 v_1 = 3.4 \times 225 = 765\text{ kg m/s}$.
For Mass 2:
Mass $m_2 = 2.5\text{ kg}$, initial speed $u_2 = 12\text{ m/s}$.
Distance in 5th second $S_{5,2} = 129\text{ m}$.
$129 = 12 + \frac{1}{2} a_2 (2 \times 5 - 1) = 12 + \frac{9}{2} a_2$.
$129 - 12 = 4.5 a_2 \implies 117 = 4.5 a_2 \implies a_2 = \frac{117}{4.5} = 26\text{ m/s}^2$.
Velocity of mass 2 after 10 seconds:
$v_2 = u_2 + a_2(10) = 12 + 26(10) = 12 + 260 = 272\text{ m/s}$.
Momentum of mass 2: $p_2 = m_2 v_2 = 2.5 \times 272 = 680\text{ kg m/s}$.
Calculate the ratio of momenta:
Ratio = $\frac{p_1}{p_2} = \frac{765}{680}$.
Divide both numerator and denominator by 85:
$\frac{765}{85} = 9$ and $\frac{680}{85} = 8$.
Ratio = $\frac{9}{8}$.
The problem states the ratio is $\frac{x}{8}$, hence $x = 9$.
Step 4: Final Answer:
The value of $x$ is 9.