A wagon weighing 1000 kg is moving with a velocity 50 km/h on smooth horizontal rails. A mass of 250 kg is dropped into it. The velocity with which it moves now is

Correct Answer: 40 km/hour

Step-by-Step Explanation:

To find the new velocity of the wagon after the mass is dropped into it, we can apply the Law of Conservation of Linear Momentum. Since the horizontal tracks are smooth and there are no external horizontal forces acting on the system, the total linear momentum in the horizontal direction remains conserved.

1. Identify the given values:
Initial mass of the wagon, $m_1=1000\text{kg}$
Initial velocity of the wagon, $v_1=50\text{km/hour}$
Mass dropped into the wagon, $m_2=250\text{kg}$ (initially has zero horizontal velocity)
Let the final velocity of the combined system be $v_2$.

2. Calculate the initial and final mass of the system:
Initial mass is just the mass of the wagon: $m_1=1000\text{kg}$.
Final mass of the system after the mass is dropped in:
$M=m_1+m_2=1000\text{kg}+250\text{kg}=1250\text{kg}$

3. Apply the conservation of linear momentum:
$\text{Initial Momentum}=\text{Final Momentum}$
$m_1\cdot v_1=(m_1+m_2)\cdot v_2$

Substitute the given values into the equation:
$1000\cdot 50=1250\cdot v_2$

Solve for the final velocity $v_2$:
$v_2=\frac{1000\cdot 50}{1250}$

Simplifying the fraction:
$v_2=\frac{50000}{1250}=40\text{km/hour}$

Thus, the velocity with which the wagon moves now is 40 km/hour.