Two carts on horizontal straight rails are pushed apart by an explosion of a powder charge Q placed between the carts. Suppose the coefficients of friction between the carts and rails are identical. If the 200 kg cart travels a distance of 36 metres and stops, the distance covered by the cart weighing 300 kg is

The correct option is 16 metres.

Let us break down the physical principles and mathematical derivations step-by-step to find the distance covered by the 300 kg cart.

Step 1: Apply the Law of Conservation of Linear Momentum
Initially, both carts are at rest, so the total initial momentum of the system is zero. When the powder charge explodes, it acts as an internal force pushing the two carts in opposite directions. Since there are no external horizontal forces acting on the system during the brief duration of the explosion, the total linear momentum must remain conserved.
Let:
- $m_1=200\text{kg}$ be the mass of the first cart,
- $m_2=300\text{kg}$ be the mass of the second cart,
- $v_1$ be the velocity of the 200 kg cart immediately after the explosion, and
- $v_2$ be the velocity of the 300 kg cart immediately after the explosion.

By conservation of momentum:
$m_1{v}_1=m_2{v}_2$
From this, we can express the ratio of their initial velocities in terms of their masses:
$\frac{v_2}{v_1}=\frac{m_1}{m_2}$
Substituting the given masses:
$\frac{v_2}{v_1}=\frac{200}{300}=\frac{2}{3}$

Step 2: Determine the Deceleration due to Friction
After the explosion, each cart moves under the influence of friction until it stops. The frictional force ($f$) opposing the motion of a cart of mass $m$ is given by:
$f=\mu mg$
where $\mu$ is the coefficient of friction and $g$ is the acceleration due to gravity.
Using Newton's second law, the deceleration ($a$) experienced by each cart is:
$a=\frac{f}{m}=\frac{\mu mg}{m}=\mu g$
Since the coefficients of friction between the carts and the rails are identical, both carts experience the exact same deceleration $a=\mu g$.

Step 3: Relate Velocity to Stopping Distance
Using the third equation of motion for a decelerating body that comes to rest (final velocity = 0):
$v^2=2ad$
where $d$ is the stopping distance. This gives:
$d=\frac{v^2}{2a}$
Since $a$ is constant for both carts, the stopping distance is directly proportional to the square of the initial velocity:
$d\propto v^2$
Therefore, the ratio of the distances covered by the two carts is:
$\frac{d_2}{d_1}={\left(\frac{v_2}{v_1}\right)}^2$

Step 4: Calculate the Distance for the 300 kg Cart
Substitute the velocity ratio derived in Step 1 into the distance ratio:
$\frac{d_2}{d_1}={\left(\frac{2}{3}\right)}^2=\frac{4}{9}$
Given that the first cart ($200\text{kg}$) travels a distance of $d_1=36\text{metres}$, we can solve for the distance $d_2$ covered by the second cart ($300\text{kg}$):
$d_2=36\times \frac{4}{9}$
$d_2=4\times 4=16\text{metres}$

Thus, the distance covered by the cart weighing 300 kg is 16 metres.