Tarzan ( $$ \mathrm{m}=85.0 \mathrm{~kg} $$ ) tries to cross a river by swinging from a $$ 10.0-\mathrm{m} $$-long vine. His speed at the bottom of the swing (as he just clears the water) is $$ 8.00 \mathrm{~m} / \mathrm{s} $$. Tarzan doesn't know that the vine has a breaking strength of $$ 1000 . \mathrm{N} $$.

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To determine whether Tarzan's vine will break as he swings across the river, we need to calculate the tension in the vine at the bottom of his swing and compare it to the vine's breaking strength.

### Given Data:
- **Mass of Tarzan**, $$ m = 85.0 \, \text{kg} $$
- **Length of the vine**, $$ L = 10.0 \, \text{m} $$
- **Speed at the bottom of the swing**, $$ v = 8.00 \, \text{m/s} $$
- **Breaking strength of the vine**, $$ T_{\text{max}} = 1000 \, \text{N} $$
- **Acceleration due to gravity**, $$ g = 9.81 \, \text{m/s}^2 $$ (standard value)

### Calculating the Tension in the Vine:
At the bottom of the swing, the tension ($$ T $$) in the vine must provide the necessary centripetal force to keep Tarzan moving in a circular path **and** support his weight. Therefore, the total tension is the sum of the centripetal force and the gravitational force.

1. **Centripetal Force ($$ F_c $$)**:
   $$
   F_c = \frac{mv^2}{L}
   $$
   Plugging in the values:
   $$
   F_c = \frac{85.0 \, \text{kg} \times (8.00 \, \text{m/s})^2}{10.0 \, \text{m}} = \frac{85.0 \times 64}{10} = \frac{5440}{10} = 544 \, \text{N}
   $$

2. **Gravitational Force ($$ F_g $$)**:
   $$
   F_g = mg
   $$
   Plugging in the values:
   $$
   F_g = 85.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 833.85 \, \text{N} \approx 834 \, \text{N}
   $$

3. **Total Tension ($$ T $$)**:
   $$
   T = F_c + F_g = 544 \, \text{N} + 834 \, \text{N} = 1378 \, \text{N}
   $$

### Comparing Tension to Breaking Strength:
- **Calculated Tension**, $$ T = 1378 \, \text{N} $$
- **Breaking Strength of the Vine**, $$ T_{\text{max}} = 1000 \, \text{N} $$

Since $$ 1378 \, \text{N} > 1000 \, \text{N} $$, the tension in the vine **exceeds** its breaking strength.

### **Conclusion:**
Tarzan's vine will **break** when he swings across the river, as the tension required to support his motion and weight surpasses the vine's maximum capacity.
The tension in the vine is 1378 N, which is greater than the vine's breaking strength of 1000 N. Therefore, the vine will break when Tarzan swings across the river.

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