A rubber ball of mass `50g` falls from a height of `100cm` and rebounds to a height of `50cm` Find the impulse and average force between the ball and the ground, if time of contact is `0.1s` .

Explanation

To solve the problem, we need to calculate the impulse and average force between the rubber ball and the ground. Let's break down the steps:

### Step 1: Convert Mass to Kilograms
The mass of the ball is given as 50 grams. We need to convert this to kilograms for standard SI units.

\[ \text{Mass} (m) = 50 \, \text{g} = 0.050 \, \text{kg} \]

**Hint:** Remember that 1 kg = 1000 g.

### Step 2: Calculate the Initial Velocity Before Impact
The ball falls from a height of 100 cm. We can use the formula for the velocity of an object in free fall:

\[ v = \sqrt{2gh} \]

Where:
- \( g \) = acceleration due to gravity = \( 9.81 \, \text{m/s}^2 \)
- \( h \) = height = \( 100 \, \text{cm} = 1.00 \, \text{m} \)

Calculating the initial velocity:

\[ v = \sqrt{2 \times 9.81 \times 1.00} = \sqrt{19.62} \approx 4.43 \, \text{m/s} \]

**Hint:** Use the square root function to find the velocity from the height.

### Step 3: Calculate the Final Velocity After Rebound
The ball rebounds to a height of 50 cm. We will use the same formula to find the velocity just after the rebound:

\[ v' = \sqrt{2gh'} \]

Where:
- \( h' = 50 \, \text{cm} = 0.50 \, \text{m} \)

Calculating the final velocity:

\[ v' = \sqrt{2 \times 9.81 \times 0.50} = \sqrt{9.81} \approx 3.13 \, \text{m/s} \]

**Hint:** Remember that the rebound velocity is positive as it moves upwards.

### Step 4: Calculate the Change in Momentum
The change in momentum (\( \Delta p \)) can be calculated using the formula:

\[ \Delta p = m(v' + v) \]

Substituting the values we have:

\[ \Delta p = 0.050 \, (3.13 + 4.43) = 0.050 \times 7.56 \approx 0.378 \, \text{kg m/s} \]

**Hint:** Note that the initial velocity is negative (downward) and the rebound velocity is positive (upward).

### Step 5: Calculate the Impulse
Impulse (\( J \)) is equal to the change in momentum:

\[ J = \Delta p \approx 0.378 \, \text{kg m/s} \]

**Hint:** Impulse is the same as the change in momentum.

### Step 6: Calculate the Average Force
The average force (\( F_{avg} \)) can be calculated using the formula:

\[ F_{avg} = \frac{J}{\Delta t} \]

Where \( \Delta t = 0.1 \, \text{s} \):

\[ F_{avg} = \frac{0.378}{0.1} = 3.78 \, \text{N} \]

**Hint:** Average force is impulse divided by the time of contact.

### Final Answers
- **Impulse:** \( \approx 0.378 \, \text{kg m/s} \)
- **Average Force:** \( \approx 3.78 \, \text{N} \)