Person A:
Weight: 1200 N
Net Impulse for counter movement jump: 240 Ns
Person B:
Weight: 800 N
Net Impulse for counter movement jump: 160 Ns
1) Which person jumped highest?
2)What was kinetic energy of each person right before they left the ground?
3)What was the potential energy of each person at apex of their jump?
4)How long was each person in the air (assuming they jumped and landed at equal heights)?
To answer these questions, we need to use the concepts of impulse, force, energy, and time. Let's break it down step by step:
1) To determine who jumped the highest, we need to compare the net impulse of their counter movement jump. Net impulse is the change in momentum and can be calculated by multiplying the average force exerted on an object by the time interval over which the force was applied. In this case, the net impulse for Person A is 240 Ns, and for Person B it is 160 Ns.
To calculate the height jumped, we can use the principle of conservation of mechanical energy. The potential energy at the highest point of the jump is equal to the initial kinetic energy before the jump. So, we need to equate the impulse to the change in kinetic energy.
The formula for impulse is:
Impulse = Force x Time
Since impulse is also equal to the change in momentum, we can write this as:
Impulse = Change in Momentum
= Mass x (Final Velocity - Initial Velocity)
In this case, since both persons are jumping vertically, we can consider the initial velocity to be zero and the final velocity to be the upward velocity achieved during the jump.
2) The kinetic energy of a person right before they leave the ground can be calculated using the formula:
Kinetic Energy = 0.5 x Mass x Velocity^2
For Person A:
Weight = 1200 N, which is equivalent to a mass of 120 kg (assuming g = 9.8 m/s^2)
Impulse = 240 Ns
Using the impulse formula, we can find the velocity achieved by Person A during the jump. Rearranging the formula:
Impulse = Mass x (Final Velocity - Initial Velocity)
240 Ns = 120 kg x (v - 0)
Therefore, Person A achieves a velocity of 2 m/s during the jump.
Substituting the values into the kinetic energy formula:
Kinetic Energy = 0.5 x 120 kg x (2 m/s)^2
Kinetic Energy = 0.5 x 120 kg x 4 m^2/s^2
Kinetic Energy = 240 J
So, Person A has a kinetic energy of 240 Joules right before leaving the ground.
Similarly, for Person B, with a weight of 800 N (equivalent to a mass of 80 kg), we can calculate the kinetic energy using the same formula:
Impulse = Mass x (Final Velocity - Initial Velocity)
160 Ns = 80 kg x (v - 0)
Person B achieves a velocity of 2 m/s during the jump.
Kinetic Energy = 0.5 x 80 kg x (2 m/s)^2
Kinetic Energy = 160 J
So, Person B has a kinetic energy of 160 Joules right before leaving the ground.
3) To determine the potential energy at the apex of the jump, we need to consider the conservation of mechanical energy. At the highest point of the jump, all of the kinetic energy turns into potential energy, neglecting any energy losses due to external forces like air resistance.
Potential Energy = Kinetic Energy
For Person A, the potential energy at the apex of the jump is 240 Joules.
For Person B, the potential energy at the apex of the jump is 160 Joules.
4) To calculate the time each person was in the air, we can use the concept of time in flight, assuming they jumped and landed at equal heights.
The time in flight can be calculated using the formula:
Time = 2 x (Vertical Displacement) / (Initial Vertical Velocity)
Since both persons landed at the same height, the vertical displacement is the same.
For Person A, the vertical displacement is equal to the height jumped, which we need to calculate using the impulse:
Impulse = Change in Momentum
240 Ns = 120 kg x (0 - v)
Rearranging the formula:
v = -2 m/s
Since the vertical displacement is related to the squared final velocity, the negative sign doesn't affect the value. Therefore, the vertical displacement for Person A is the same as Person B.
For Person B, with a weight of 800 N (equivalent to a mass of 80 kg), we can calculate the height of the jump using the impulse:
Impulse = Change in Momentum
160 Ns = 80 kg x (0 - v)
v = -2 m/s
The vertical displacement for Person B is also equal to Person A.
Substituting the values into the time formula:
Time = 2 x (Vertical Displacement) / (Initial Vertical Velocity)
Time = 2 x (Vertical Displacement) / (-2 m/s)
Since the vertical displacement is the same for both Person A and Person B, the time in air will also be the same.
To summarize:
1) It cannot be determined who jumped the highest based on the given information, as it depends on factors such as technique and muscular power.
2) Person A has a kinetic energy of 240 Joules, and Person B has a kinetic energy of 160 Joules right before leaving the ground.
3) Person A has a potential energy of 240 Joules, and Person B has a potential energy of 160 Joules at the apex of their jump.
4) Assuming they jumped and landed at equal heights, both Person A and Person B were in the air for the same amount of time.