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A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.00 m/ s at the bottom, what is the height of the hill?

A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.00 m/ s at the bottom, what is the height of the hill?

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I am getting stuck at the arithmetic part. Please let me know the steps in detail. TIA

### Example Physics Problem: Kinetic and Potential Energy **Problem P.5.33** A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.00 m/s at the bottom, what is the height of the hill? --- **Explanation:** Given data: - Combined mass (m) = 50.0 kg - Final speed (v) = 3.00 m/s - Initial speed (u) = 0 m/s (starts from rest) - Acceleration due to gravity (g) = 9.8 m/s² (standard value) To find: Height of the hill (h) ### Solution: Using the principle of conservation of energy, the potential energy lost by the sled-child system as it descends the hill is converted to kinetic energy. At the top of the hill (initial position): - Kinetic Energy (KE_initial) = 0 (because the speed is 0) - Potential Energy (PE_initial) = mgh At the bottom of the hill (final position): - Kinetic Energy (KE_final) = 0.5 * m * v² - Potential Energy (PE_final) = 0 (since the height is 0) According to the conservation of energy: \[ PE_{\text{initial}} = KE_{\text{final}} \] So, \[ mgh = \frac{1}{2} mv^2 \] Simplifying, \[ gh = \frac{1}{2} v^2 \] Therefore, solving for h: \[ h = \frac{v^2}{2g} \] Substituting the known values: \[ h = \frac{(3.00 \, \text{m/s})^2}{2 \times 9.8 \, \text{m/s}^2} \] \[ h = \frac{9.00 \, \text{m}^2/\text{s}^2}{19.6 \, \text{m/s}^2} \] \[ h = 0.459 \, \text{m} \] So, the height of the hill is approximately 0.459 meters. This example demonstrates the application of the conservation of energy principle in physics to determine the height of a hill based on the mass and final velocity of
Transcribed Image Text:### Example Physics Problem: Kinetic and Potential Energy **Problem P.5.33** A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.00 m/s at the bottom, what is the height of the hill? --- **Explanation:** Given data: - Combined mass (m) = 50.0 kg - Final speed (v) = 3.00 m/s - Initial speed (u) = 0 m/s (starts from rest) - Acceleration due to gravity (g) = 9.8 m/s² (standard value) To find: Height of the hill (h) ### Solution: Using the principle of conservation of energy, the potential energy lost by the sled-child system as it descends the hill is converted to kinetic energy. At the top of the hill (initial position): - Kinetic Energy (KE_initial) = 0 (because the speed is 0) - Potential Energy (PE_initial) = mgh At the bottom of the hill (final position): - Kinetic Energy (KE_final) = 0.5 * m * v² - Potential Energy (PE_final) = 0 (since the height is 0) According to the conservation of energy: \[ PE_{\text{initial}} = KE_{\text{final}} \] So, \[ mgh = \frac{1}{2} mv^2 \] Simplifying, \[ gh = \frac{1}{2} v^2 \] Therefore, solving for h: \[ h = \frac{v^2}{2g} \] Substituting the known values: \[ h = \frac{(3.00 \, \text{m/s})^2}{2 \times 9.8 \, \text{m/s}^2} \] \[ h = \frac{9.00 \, \text{m}^2/\text{s}^2}{19.6 \, \text{m/s}^2} \] \[ h = 0.459 \, \text{m} \] So, the height of the hill is approximately 0.459 meters. This example demonstrates the application of the conservation of energy principle in physics to determine the height of a hill based on the mass and final velocity of
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