A boy of mass 25 kg stands on a board of maas 10 kg which in turn is kept on a frictionless horizontal ice surface. The boy maks a jump with a velocity component 5m/s in a horizontal direction with respect to the ice. With what velocity does the board recoil? with what rate are the boy and the board seperating from each other?

Explanation

To solve the problem step by step, we will apply the principle of conservation of momentum.

### Step 1: Identify the masses and initial conditions
- Mass of the boy (m1) = 25 kg
- Mass of the board (m2) = 10 kg
- Initial velocity of both the boy and the board = 0 m/s (since they are at rest)

### Step 2: Write the equation for conservation of momentum
According to the law of conservation of momentum, the total momentum before an event must equal the total momentum after the event.

Initial momentum (Pi) = Final momentum (Pf)

Since both the boy and the board are initially at rest:
\[ Pi = 0 \]

### Step 3: Write the expression for final momentum
After the boy jumps, the final momentum can be expressed as:
\[ Pf = m1 \cdot v1 + m2 \cdot v2 \]
Where:
- \( v1 \) = velocity of the boy after jumping = 5 m/s (in the horizontal direction)
- \( v2 \) = velocity of the board (which we need to find)

### Step 4: Set up the equation using conservation of momentum
Since the initial momentum is zero, we have:
\[ 0 = (m1 \cdot v1) + (m2 \cdot v2) \]

Substituting the known values:
\[ 0 = (25 \cdot 5) + (10 \cdot v2) \]

### Step 5: Solve for the velocity of the board (v2)
Rearranging the equation:
\[ 10 \cdot v2 = - (25 \cdot 5) \]
\[ 10 \cdot v2 = -125 \]
\[ v2 = -12.5 \, \text{m/s} \]

The negative sign indicates that the board moves in the opposite direction to the boy's jump.

### Step 6: Calculate the rate at which the boy and the board are separating
The relative velocity (Vr) between the boy and the board can be calculated as:
\[ Vr = |v1| + |v2| \]
Since \( v1 = 5 \, \text{m/s} \) (in the positive direction) and \( v2 = -12.5 \, \text{m/s} \) (in the negative direction):
\[ Vr = 5 + 12.5 = 17.5 \, \text{m/s} \]

### Final Answers
- The velocity of the board (v2) = -12.5 m/s (indicating it moves in the opposite direction)
- The rate at which the boy and the board are separating = 17.5 m/s