A 42-kg pole vaulter running at 14 m/s vaults over the bar. Her speed when she is above the bar is 1.7 m/s. Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she crosses the bar.
KE at beginning: 1/2 m 14^2
KE at top: 1/2 m 1.7^2
Ke at beginning=KE at top+PE at top
1/2 m 14^2=1/2 m 1.7^2 + mgh
solve for h.
9.853m
To determine the altitude of the pole vaulter as she crosses the bar, we can use the law of conservation of mechanical energy.
The mechanical energy of the pole vaulter can be calculated using the equation:
E = KE + PE
Where:
E is the mechanical energy
KE is the kinetic energy
PE is the potential energy
Initially, when the pole vaulter is running before vaulting, her kinetic energy (KE) can be calculated as:
KE = 1/2 * m * v^2
Where:
m is the mass of the pole vaulter (42 kg)
v is the velocity of the pole vaulter (14 m/s)
KE = 1/2 * 42 kg * (14 m/s)^2
KE = 1/2 * 42 kg * 196 m^2/s^2
KE = 4116 J
At the maximum point of her jump, her speed decreases to 1.7 m/s. At this point, her kinetic energy (KE) can be calculated as:
KE = 1/2 * m * v^2
Where:
m is the mass of the pole vaulter (42 kg)
v is the velocity of the pole vaulter (1.7 m/s)
KE = 1/2 * 42 kg * (1.7 m/s)^2
KE = 1/2 * 42 kg * 2.89 m^2/s^2
KE = 61.17 J
Since there is no energy absorbed by the pole and no air resistance, the total mechanical energy (E) is conserved. Therefore, the potential energy (PE) at the maximum point of her jump can be calculated as:
E = KE + PE
Since the kinetic energy decreases from 4116 J to 61.17 J, the potential energy must increase by the same amount.
PE = E - KE
PE = 4116 J - 61.17 J
PE = 4054.83 J
The potential energy (PE) is equal to the gravitational potential energy (PE = m * g * h), where g is the acceleration due to gravity (9.8 m/s^2) and h is the altitude above the bar.
PE = m * g * h
4054.83 J = 42 kg * 9.8 m/s^2 * h
4054.83 J = 411.6 kg·m^2/s^2 * h
h = 4054.83 J / (411.6 kg·m^2/s^2)
h ≈ 9.846 m
Therefore, the altitude of the pole vaulter as she crosses the bar is approximately 9.846 meters.
To determine the pole vaulter's altitude as she crosses the bar, we can use the principle of conservation of mechanical energy. The mechanical energy of the vaulter consists of two components: kinetic energy (KE) due to her velocity and potential energy (PE) due to her height.
The total mechanical energy (E) of the pole vaulter is given by:
E = KE + PE
Initially, when the vaulter is running on the ground, the potential energy is zero since her height is zero. Therefore, the mechanical energy is equal to the kinetic energy:
E_initial = KE_initial
We can calculate the initial kinetic energy using the formula:
KE_initial = 1/2 * mass * velocity^2
Where:
mass = 42 kg (mass of the vaulter)
velocity = 14 m/s (velocity of the vaulter)
Substituting the values into the formula:
KE_initial = 1/2 * 42 kg * (14 m/s)^2
= 1/2 * 42 kg * 196 m^2/s^2
= 4116 kg m^2/s^2
Now, as the vaulter crosses the bar, her potential energy increases while her kinetic energy decreases because her velocity decreases. The final mechanical energy is given by:
E_final = KE_final + PE_final
Since air resistance and energy absorbed by the pole are neglected, we can assume that the total mechanical energy is conserved. Therefore, E_final is equal to E_initial:
E_final = E_initial
We can calculate the final kinetic energy using the formula:
KE_final = 1/2 * mass * velocity_final^2
Where:
mass = 42 kg (same as before)
velocity_final = 1.7 m/s (velocity of the vaulter above the bar)
Substituting the values into the formula:
KE_final = 1/2 * 42 kg * (1.7 m/s)^2
= 1/2 * 42 kg * 2.89 m^2/s^2
= 60.57 kg m^2/s^2
Now, we can rewrite the conservation of mechanical energy equation:
E_final = E_initial
KE_final + PE_final = KE_initial
Substituting the values:
60.57 kg m^2/s^2 + PE_final = 4116 kg m^2/s^2
Now, we can solve for the potential energy (PE_final):
PE_final = 4116 kg m^2/s^2 - 60.57 kg m^2/s^2
PE_final = 4055.43 kg m^2/s^2
Therefore, the vaulter's altitude as she crosses the bar is 4055.43 kg m^2/s^2.