A boy of mass 50 kg running at 5 m/s jumps on to a 20 kg trolley travelling in the same direction at 1.5 m/s. What is their common velocity ?

Answer

To solve the problem of finding the common velocity of a boy and a trolley after the boy jumps onto the trolley, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and velocities
- Mass of the boy (m1) = 50 kg
- Velocity of the boy (v1) = 5 m/s
- Mass of the trolley (m2) = 20 kg
- Velocity of the trolley (v2) = 1.5 m/s

### Step 2: Calculate the total momentum before the collision
The total momentum before the boy jumps onto the trolley can be calculated using the formula for momentum (p = mass × velocity).

- Momentum of the boy = m1 × v1 = 50 kg × 5 m/s = 250 kg·m/s
- Momentum of the trolley = m2 × v2 = 20 kg × 1.5 m/s = 30 kg·m/s

Now, add the two momenta together to get the total momentum before the collision:
\[ \text{Total momentum before} = \text{Momentum of boy} + \text{Momentum of trolley} \]
\[ \text{Total momentum before} = 250 kg·m/s + 30 kg·m/s = 280 kg·m/s \]

### Step 3: Calculate the total mass after the collision
After the boy jumps onto the trolley, the total mass of the system (boy + trolley) is:
\[ \text{Total mass} = m1 + m2 = 50 kg + 20 kg = 70 kg \]

### Step 4: Set up the equation for momentum after the collision
Let the common velocity after the boy jumps onto the trolley be \( v \). The total momentum after the collision can be expressed as:
\[ \text{Total momentum after} = \text{Total mass} \times v = 70 kg \times v \]

### Step 5: Apply the law of conservation of momentum
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
\[ \text{Total momentum before} = \text{Total momentum after} \]
\[ 280 kg·m/s = 70 kg \times v \]

### Step 6: Solve for the common velocity \( v \)
To find \( v \), rearrange the equation:
\[ v = \frac{280 kg·m/s}{70 kg} \]
\[ v = 4 m/s \]

### Conclusion
The common velocity of the boy and the trolley after the boy jumps onto the trolley is **4 m/s**.

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