To determine the additional distance the weights contribute to the long jumper’s range, we apply the principle of conservation of momentum and projectile motion.

Step 1: Define the Given Values

• Mass of the jumper + weights: $M = 75 + 2(5.1) = 85.2$ kg
• Mass of the jumper after throwing weights: $m_j = 75$ kg
• Mass of each weight: $m_w = 5.1$ kg
• Initial velocity: $\mathbf{v}_i = (9.3 \hat{i} + 4.1 \hat{j})$ m/s
• Final velocity of weights: $\mathbf{v}_w = 0$ (since they stop moving relative to the ground)

Step 2: Apply Conservation of Momentum (Horizontal)

Since no external horizontal forces act on the system, the total momentum before and after throwing the weights must remain the same.

Before Throwing the Weights:

Total horizontal momentum:

$P_{\text{initial}} = M v_{ix}$

$P_{\text{initial}} = (85.2)(9.3)$

$P_{\text{initial}} = 792.36 \text{ kg·m/s}$

After Throwing the Weights:

The weights come to rest relative to the ground, so they contribute no horizontal momentum. The jumper’s new horizontal velocity $v_{fx}$ satisfies:

$m_j v_{fx} = P_{\text{initial}}$

$75 v_{fx} = 792.36$

$v_{fx} = \frac{792.36}{75} = 10.565 \text{ m/s}$

Step 3: Calculate Additional Distance

Since the time of flight remains unchanged (because vertical motion is unaffected), we use the standard kinematic equation for projectile motion:

$t = \frac{2 v_{iy}}{g}$

$t = \frac{2(4.1)}{9.81}$

$t = \frac{8.2}{9.81} \approx 0.836 \text{ s}$

The additional horizontal distance covered due to the velocity increase:

$\Delta x = (v_{fx} - v_{ix}) t$

$\Delta x = (10.565 - 9.3)(0.836)$

$\Delta x = (1.265)(0.836)$

$\Delta x \approx 1.06 \text{ m} = \mathbf{106} \text{ cm}$

Final Answer:

The weights add 106 cm to the jumper’s range.

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Related Questions:

1. How does the time of flight remain the same in this scenario?
2. What if the weights were thrown at an angle instead of horizontally?
3. How would air resistance affect the jumper's range?
4. What happens if the jumper throws only one weight?
5. How much additional distance would be gained if the jumper were lighter?

Tip:

Always check whether an external force (like air resistance) might impact conservation of momentum before assuming it's conserved. 🚀