Question

A block moving horizontally on a smooth surface with a speed of 40 m/s splits into two parts with masses in the ratio of \(1 : 2\). If the smaller part moves at 60 m/s in the same direction, then the fractional change in kinetic energy is :

• A. \(\frac{1}{8}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{2}{3}\)

Hint:

In explosion/splitting problems, the kinetic energy of the system always increases (\(\Delta K>0\)) because chemical or internal potential energy is converted into kinetic energy. If you get a negative value, recheck your momentum conservation calculation.

Answer 1

The Correct Option is A

To find the fractional change in kinetic energy, we need to use the concepts of conservation of momentum and kinetic energy. Here's how we solve the problem step-by-step:

1. Initial conditions:
   • Initial velocity of the block v = 40\,\text{m/s}
   • Total initial mass of the block m
2. Given the block splits into two parts with a mass ratio of 1:2, let's denote:
   • Mass of smaller part m_1 = \frac{m}{3}
   • Mass of larger part m_2 = \frac{2m}{3}
3. The smaller part moves with a velocity v_1 = 60\,\text{m/s} in the same direction.
4. Using the conservation of linear momentum, the momentum before and after the split should be equal: mv = m_1v_1 + m_2v_2
5. Substitute the known values into the momentum equation: m \times 40 = \left(\frac{m}{3}\right) \times 60 + \left(\frac{2m}{3}\right) \times v_2
6. Solve for v_2:
   • Calculate 60m/3 = 20m
   • Simplify to: 40m = 20m + \frac{2mv_2}{3}
   • Rearrange: 20m = \frac{2mv_2}{3}
   • Multiplying by 3/2: v_2 = 30\,\text{m/s}
7. Calculate the initial kinetic energy (\text{KE}_\text{initial}): \text{KE}_\text{initial} = \frac{1}{2}mv^2 = \frac{1}{2}m(40)^2 = 800m
8. Calculate the final kinetic energy (\text{KE}_\text{final}): \text{KE}_\text{final} = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2
9. Substitute the known values:
   • m_1v_1^2 = \frac{m}{3} \times 60^2 = 1200m
   • m_2v_2^2 = \frac{2m}{3} \times 30^2 = 600m
   • \text{KE}_\text{final} = \frac{1}{2}(1200m + 600m) = 900m
10. Calculate the change in kinetic energy: \Delta \text{KE} = \text{KE}_\text{final} - \text{KE}_\text{initial} = 900m - 800m = 100m
11. Find the fractional change in kinetic energy: \frac{\Delta \text{KE}}{\text{KE}_\text{initial}} = \frac{100m}{800m} = \frac{1}{8}

Thus, the fractional change in kinetic energy is \frac{1}{8}.