To determine the altitude the pole vaulter reaches as she crosses the bar, we can apply the principle of conservation of mechanical energy. The initial kinetic energy of the pole vaulter when she is running can be converted into kinetic energy when she is at the peak of her jump, plus the gravitational potential energy at that height.
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Initial kinetic energy (KE_initial) when running: \[ KE_{\text{initial}} = \frac{1}{2} m v^2 \] where \(m = 68 , \text{kg}\) and \(v = 10 , \text{m/s}\).
\[ KE_{\text{initial}} = \frac{1}{2} \cdot 68 \cdot (10)^2 = \frac{1}{2} \cdot 68 \cdot 100 = 3400 , \text{J} \]
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Final kinetic energy (KE_final) as she crosses the bar with speed 1.5 m/s: \[ KE_{\text{final}} = \frac{1}{2} m v^2 \] where \(v = 1.5 , \text{m/s}\).
\[ KE_{\text{final}} = \frac{1}{2} \cdot 68 \cdot (1.5)^2 = \frac{1}{2} \cdot 68 \cdot 2.25 = 76.5 , \text{J} \]
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Conservation of energy tells us that the initial kinetic energy minus the final kinetic energy equals the gravitational potential energy (PE) at the height \(h\): \[ KE_{\text{initial}} - KE_{\text{final}} = PE \] \[ 3400 , \text{J} - 76.5 , \text{J} = mgh \] where \(g = 9.81 , \text{m/s}^2\).
Rearranging it gives: \[ 3400 - 76.5 = 68 \cdot 9.81 \cdot h \] \[ 3323.5 = 68 \cdot 9.81 \cdot h \] \[ h = \frac{3323.5}{68 \cdot 9.81} \]
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Calculating \(h\): \[ h = \frac{3323.5}{666.68} \approx 4.98 , \text{m} \]
Thus, the altitude as the pole vaulter crosses the bar is approximately 4.98 meters.
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