A man of mass `60kg` jumps from a trolley of mass `20kg` standing on smooth surface with absolute velocity `3m//s`. Find velocity of trolley and total energy produced by man.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the System
We have a man with a mass of \( m_1 = 60 \, \text{kg} \) and a trolley with a mass of \( m_2 = 20 \, \text{kg} \). The man jumps off the trolley with an absolute velocity of \( v_{m} = 3 \, \text{m/s} \). The initial momentum of the system is zero because both the man and the trolley are initially at rest.

### Step 2: Apply Conservation of Momentum
Since there are no external forces acting on the system, we can apply the principle of conservation of linear momentum. The initial momentum of the system is given by:

\[ \text{Initial Momentum} = 0 \]

After the man jumps off, the final momentum of the system can be expressed as:

\[ \text{Final Momentum} = m_1 \cdot v_{m} + m_2 \cdot (-v_{t}) \]

Where \( v_{t} \) is the velocity of the trolley (in the opposite direction, hence the negative sign). Setting the initial momentum equal to the final momentum gives us:

\[ 0 = (60 \, \text{kg}) \cdot (3 \, \text{m/s}) + (20 \, \text{kg}) \cdot (-v_{t}) \]

### Step 3: Solve for the Velocity of the Trolley
Rearranging the equation to solve for \( v_{t} \):

\[ 60 \cdot 3 = 20 \cdot v_{t} \]

\[ 180 = 20 \cdot v_{t} \]

\[ v_{t} = \frac{180}{20} = 9 \, \text{m/s} \]

Thus, the velocity of the trolley is \( v_{t} = 9 \, \text{m/s} \) in the backward direction.

### Step 4: Calculate the Kinetic Energy
Next, we need to calculate the total kinetic energy produced by the man after he jumps off. The kinetic energy (KE) is given by the formula:

\[ KE = \frac{1}{2} m v^2 \]

**For the man:**
\[ KE_{man} = \frac{1}{2} \cdot 60 \, \text{kg} \cdot (3 \, \text{m/s})^2 = \frac{1}{2} \cdot 60 \cdot 9 = 270 \, \text{J} \]

**For the trolley:**
\[ KE_{trolley} = \frac{1}{2} \cdot 20 \, \text{kg} \cdot (9 \, \text{m/s})^2 = \frac{1}{2} \cdot 20 \cdot 81 = 810 \, \text{J} \]

### Step 5: Total Kinetic Energy
Now, we can find the total kinetic energy after the jump:

\[ KE_{total} = KE_{man} + KE_{trolley} = 270 \, \text{J} + 810 \, \text{J} = 1080 \, \text{J} \]

### Step 6: Calculate the Energy Produced by the Man
The energy produced by the man is equal to the change in kinetic energy. Since the initial kinetic energy was zero (both were at rest), the energy produced by the man is:

\[ \text{Energy produced} = KE_{total} - KE_{initial} = 1080 \, \text{J} - 0 \, \text{J} = 1080 \, \text{J} \]

### Final Answers
- The velocity of the trolley is \( 9 \, \text{m/s} \) backward.
- The total energy produced by the man is \( 1080 \, \text{J} \).

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