A 54 kg pole vaulter running at 10 m/s vaults over the bar. Her speed when she is above the bar is 1.3 m/s. The acceleration of gravity is 9.8 m/s 2 . Find her altitude as she crosses the bar. Neglect air resistance, as well as any energy absorbed by the pole. Answer in units of m
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To find the altitude of the pole vaulter as she crosses the bar, we can use the principle of conservation of energy. This principle states that the total mechanical energy (kinetic energy + potential energy) is conserved in an isolated system, neglecting air resistance and other non-conservative forces.
Initial Situation (When the pole vaulter is running at full speed before the jump): - Kinetic Energy (KE_initial) = 1/2 * m * v^2, where m is the mass of the pole vaulter, and v is her speed before the jump.
Final Situation (When the pole vaulter is above the bar): - Kinetic Energy (KE_final) = 1/2 * m * v'^2, where v' is her speed above the bar. - Potential Energy (PE_final) = m * g * h, where g is the acceleration due to gravity, and h is the height above the ground (or the altitude as she crosses the bar).
Since we neglect energy absorbed by the pole and air resistance, the total mechanical energy remains constant, so:
KE_initial + PE_initial = KE_final + PE_final
Considering that the pole vaulter starts from the ground, her initial potential energy (PE_initial) is 0. Therefore, we have:
1/2 * m * v^2 = 1/2 * m * v'^2 + m * g * h
Since we are looking for the altitude 'h', we can solve for 'h':
h = [(1/2 * m * v^2) - (1/2 * m * v'^2)] / (m * g)
Notice that the mass 'm' cancels out, simplifying our equation to:
h = [(1/2 * v^2) - (1/2 * v'^2)] / g
Now, we can plug in the values given in the problem:
v = 10 m/s (initial speed) v' = 1.3 m/s (speed above the bar) g = 9.8 m/s² (acceleration due to gravity)
h = [(1/2 * (10)^2) - (1/2 * (1.3)^2)] / 9.8
h = [(1/2 * 100) - (1/2 * 1.69)] / 9.8
h = (50 - 0.845) / 9.8 h = 49.155 / 9.8 h ≈ 5.017 m
So, the altitude of the pole vaulter as she crosses the bar is approximately 5.017 meters.