To solve the problem step by step, we will use the principle of conservation of momentum.
### Step 1: Convert the given velocity from km/h to m/s
The velocity of Ishan is given as \(6.75 \, \text{km/h}\). We need to convert this to meters per second (m/s) using the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\]
Thus, we calculate:
\[
V = 6.75 \times \frac{5}{18} = 1.875 \, \text{m/s}
\]
### Step 2: Convert the mass of the trolley from quintals to kilograms
The mass of the trolley is given as \(0.25 \, \text{quintal}\). We know that:
\[
1 \, \text{quintal} = 100 \, \text{kg}
\]
Therefore, we convert the mass of the trolley:
\[
\text{Mass of trolley} = 0.25 \times 100 = 25 \, \text{kg}
\]
### Step 3: Apply the conservation of momentum
Before Ishan jumps into the trolley, the momentum of the system is:
\[
\text{Initial momentum} = \text{momentum of Ishan} + \text{momentum of trolley}
\]
\[
= (50 \, \text{kg} \times 1.875 \, \text{m/s}) + (25 \, \text{kg} \times 0 \, \text{m/s}) = 93.75 \, \text{kg m/s}
\]
After Ishan jumps into the trolley, both Ishan and the trolley move together with a common velocity \(V_f\). The total mass of the system is:
\[
\text{Total mass} = 50 \, \text{kg} + 25 \, \text{kg} = 75 \, \text{kg}
\]
The final momentum of the system is:
\[
\text{Final momentum} = \text{Total mass} \times V_f = 75 \, \text{kg} \times V_f
\]
### Step 4: Set initial momentum equal to final momentum
Using the conservation of momentum:
\[
93.75 \, \text{kg m/s} = 75 \, \text{kg} \times V_f
\]
### Step 5: Solve for \(V_f\)
Now we can solve for \(V_f\):
\[
V_f = \frac{93.75 \, \text{kg m/s}}{75 \, \text{kg}} = 1.25 \, \text{m/s}
\]
### Final Answer
The velocity with which the trolley starts moving along the rails is:
\[
\boxed{1.25 \, \text{m/s}}
\]
