A 75.0 kg person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of 13.5 cm. Model the floor as completely solid and motionless.
To model the motion of the person jumping, we can use the principles of conservation of energy. The person starts with gravitational potential energy when standing on the floor and converts it into kinetic energy as they jump up.
First, let's calculate the gravitational potential energy that the person initially has. The formula for gravitational potential energy is given by:
PE = mgh
Where:
PE is the potential energy,
m is the mass of the person (75.0 kg),
g is the acceleration due to gravity (9.8 m/s²),
h is the maximum height the person reaches (13.5 cm or 0.135 m).
PE = (75.0 kg) * (9.8 m/s²) * (0.135 m)
PE = 122.03 Joules
This initial potential energy gets converted into the person's maximum height, where all the potential energy is converted into kinetic energy. At the maximum height, all the potential energy is converted into kinetic energy.
KE = PE
The formula for kinetic energy is given by:
KE = 1/2 * m * v²
Where:
KE is the kinetic energy,
m is the mass of the person (75.0 kg),
v is the velocity of the person when reaching the maximum height (which is 0 since the person momentarily stops at the top).
So, at the maximum height, we can equate the kinetic energy to the potential energy:
1/2 * m * v² = PE
Simplifying:
1/2 * (75.0 kg) * v² = 122.03 Joules
Solve for v²:
v² = (2 * 122.03 Joules) / (75.0 kg)
v² = 2.61 m²/s²
Take the square root to find v:
v ≈ 1.61 m/s
The velocity of the person at the maximum height is approximately 1.61 m/s.