Activity 4: Change in momentum ( 40 minutes) 1 In each of the following cases: - Calculate the magnitude and direction of the ínitial momentum, final momentum and change in momentum of the object. - Draw a labelled vector diagram that shows the initial and final momentum of the object as well as its change in momentum. 1.1 A 1100 kg car accelerates along a straight road and its velocity increases from $$ 10 \mathrm{~m} . \mathrm{s}^{-1} $$ to $$ 20 \mathrm{~m} . \mathrm{s}^{-1} $$. 1.2 The brakes of a 1500 kg car are applied and its velocity decreases from $$ 10 \mathrm{~m}_{\mathrm{s}} \mathrm{s}^{-1} $$ to $$ 2 \mathrm{~m} \cdot \mathrm{~s}^{-1} $$. 1.3 A 440 g soccer ball approaches a goalkeeper at $$ 8 \mathrm{~m} . \mathrm{s}^{-1} $$ and he kicks it in the opposite direction at $$ 10 \mathrm{~m} \cdot \mathrm{~s}^{-1} $$

Question

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To solve the problem, we will follow these steps for each case:

1. Calculate the initial momentum ($$ p_i $$), final momentum ($$ p_f $$), and change in momentum ($$ \Delta p $$).
2. Draw a labeled vector diagram to illustrate the initial momentum, final momentum, and change in momentum.

The formula for momentum is given by:
$$
p = m \cdot v
$$
where $$ p $$ is momentum, $$ m $$ is mass, and $$ v $$ is velocity.

### Case 1.1: Car Accelerating

**Given:**
- Mass of the car, $$ m = 1100 \, \text{kg} $$
- Initial velocity, $$ v_i = 10 \, \text{m/s} $$
- Final velocity, $$ v_f = 20 \, \text{m/s} $$

**Calculations:**
1. **Initial Momentum:**
   $$
   p_i = m \cdot v_i = 1100 \, \text{kg} \cdot 10 \, \text{m/s} = 11000 \, \text{kg m/s}
   $$

2. **Final Momentum:**
   $$
   p_f = m \cdot v_f = 1100 \, \text{kg} \cdot 20 \, \text{m/s} = 22000 \, \text{kg m/s}
   $$

3. **Change in Momentum:**
   $$
   \Delta p = p_f - p_i = 22000 \, \text{kg m/s} - 11000 \, \text{kg m/s} = 11000 \, \text{kg m/s}
   $$

### Case 1.2: Car Braking

**Given:**
- Mass of the car, $$ m = 1500 \, \text{kg} $$
- Initial velocity, $$ v_i = 10 \, \text{m/s} $$
- Final velocity, $$ v_f = 2 \, \text{m/s} $$

**Calculations:**
1. **Initial Momentum:**
   $$
   p_i = m \cdot v_i = 1500 \, \text{kg} \cdot 10 \, \text{m/s} = 15000 \, \text{kg m/s}
   $$

2. **Final Momentum:**
   $$
   p_f = m \cdot v_f = 1500 \, \text{kg} \cdot 2 \, \text{m/s} = 3000 \, \text{kg m/s}
   $$

3. **Change in Momentum:**
   $$
   \Delta p = p_f - p_i = 3000 \, \text{kg m/s} - 15000 \, \text{kg m/s} = -12000 \, \text{kg m/s}
   $$

### Case 1.3: Soccer Ball Kicked

**Given:**
- Mass of the soccer ball, $$ m = 440 \, \text{g} = 0.440 \, \text{kg} $$
- Initial velocity, $$ v_i = 8 \, \text{m/s} $$ (approaching)
- Final velocity, $$ v_f = -10 \, \text{m/s} $$ (kicked in the opposite direction)

**Calculations:**
1. **Initial Momentum:**
   $$
   p_i = m \cdot v_i = 0.440 \, \text{kg} \cdot 8 \, \text{m/s} = 3.52 \, \text{kg m/s}
   $$

2. **Final Momentum:**
   $$
   p_f = m \cdot v_f = 0.440 \, \text{kg} \cdot (-10) \, \text{m/s} = -4.4 \, \text{kg m/s}
   $$

3. **Change in Momentum:**
   $$
   \Delta p = p_f - p_i = -4.4 \, \text{kg m/s} - 3.52 \, \text{kg m/s} = -7.92 \, \text{kg m/s}
   $$

### Summary of Results

1. **Case 1.1:**
   - Initial Momentum: $$ 11000 \, \text{kg m/s} $$ (forward)
   - Final Momentum: $$ 22000 \, \text{kg m/s} $$ (forward)
   - Change in Momentum: $$ 11000 \, \text{kg m/s} $$ (forward)

2. **Case 1.2:**
   - Initial Momentum: $$ 15000 \, \text{kg m/s} $$ (forward)
   - Final Momentum: $$ 3000 \, \text{kg m/s} $$ (forward)
   - Change in Momentum: $$ -12000 \, \text{kg m/s} $$ (backward)

3. **Case 1.3:**
   - Initial Momentum: $$ 3.52 \, \text{kg m/s} $$ (forward)
   - Final Momentum: $$ -4.4 \, \text{kg m/s} $$ (backward)
   - Change in Momentum: $$ -7.92 \, \text{kg m/s} $$ (backward)

### Vector Diagrams
For each case, you would draw a vector diagram showing the initial momentum vector pointing in the direction of motion, the final momentum vector also pointing in the direction of motion (or opposite for the soccer ball), and the change in momentum vector which would be the difference between the final and initial momentum vectors.

If you need assistance with drawing the vector diagrams, please let me know!
**Case 1.1:** - Initial Momentum: 11,000 kg·m/s (forward) - Final Momentum: 22,000 kg·m/s (forward) - Change in Momentum: 11,000 kg·m/s (forward) **Case 1.2:** - Initial Momentum: 15,000 kg·m/s (forward) - Final Momentum: 3,000 kg·m/s (forward) - Change in Momentum: -12,000 kg·m/s (backward) **Case 1.3:** - Initial Momentum: 3.52 kg·m/s (forward) - Final Momentum: -4.4 kg·m/s (backward) - Change in Momentum: -7.92 kg·m/s (backward) For each case, draw a vector diagram showing the initial and final momentum vectors and the change in momentum vector.

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