Two people jump out of an airplane. They hold on to each other while falling straight down at a shared terminal speed of 60.7 m/s. Suddenly, they push away from each other. Immediately after separation, the first skydiver (who has a mass of 89.3 kg) has the following velocity components (with straight down corresponding to the positive z-axis):
v1x: 4.43 m/s
v1y: 4.75 m/s
v1z: 60.7 m/s
What are the x- and y-components of the velocity of the second skydiver, whose mass is 57.7 kg, immediately after separation?
What is the change in kinetic energy of the system?
To find the x- and y-components of the velocity of the second skydiver immediately after separation, we can use the conservation of momentum. The total momentum of the system before separation should be equal to the total momentum after separation.
Let's assume the x- and y-components of the velocity of the second skydiver are v2x and v2y, respectively.
The total momentum in the x-direction before separation:
P1x = m1 * v1x = (89.3 kg) * (4.43 m/s) = 395.199 kg*m/s
The total momentum in the y-direction before separation:
P1y = m1 * v1y = (89.3 kg) * (4.75 m/s) = 424.675 kg*m/s
The total momentum in the x-direction after separation:
P2x = m2 * v2x
The total momentum in the y-direction after separation:
P2y = m2 * v2y
Since the total momentum before separation should be equal to the total momentum after separation, we have:
P1x + P1y = P2x + P2y
395.199 kg*m/s + 424.675 kg*m/s = (m2 * v2x) + (m2 * v2y)
Since the skydivers are falling straight down at a shared terminal speed of 60.7 m/s, the z-component of the velocity for both skydivers should be the same. Therefore, the z-component of the velocity for the second skydiver should also be 60.7 m/s.
Next, we need to find the change in kinetic energy of the system after separation. Kinetic energy is given by the equation:
KE = (1/2) * m * v^2
The change in kinetic energy can be calculated as the difference between the final and initial kinetic energies of the system.
The initial kinetic energy of the system can be calculated as:
KE1 = (1/2) * (m1 * v1^2) + (m2 * v1^2)
The final kinetic energy of the system can be calculated as:
KE2 = (1/2) * (m1 * v1^2) + (m2 * (v2^2 + v2^2 + v2^2))
The change in kinetic energy of the system is then:
ΔKE = KE2 - KE1
To find the x- and y-components of the velocity of the second skydiver, we first need to solve the momentum equation and find the values of v2x and v2y. However, with the given information, it is not possible to determine these values.
To find the x- and y-components of the velocity of the second skydiver, we can use the principle of conservation of momentum. According to this principle, the total momentum before separation should be equal to the total momentum after separation.
The total momentum before separation is given by the sum of the individual momenta:
Total Momentum Before = Momentum of Skydiver 1 + Momentum of Skydiver 2
Since they are falling straight down (along the positive z-axis) and their masses are known, we can use the formula for momentum (p = mv) to calculate their individual momenta:
Momentum of Skydiver 1 = mass of Skydiver 1 * velocity of Skydiver 1
= 89.3 kg * (4.43 m/s * i + 4.75 m/s * j + 60.7 m/s * k)
Since they push away from each other, the total momentum after separation is the vector sum of the individual momenta in the x- and y-directions:
Total Momentum After = Momentum of Skydiver 1' + Momentum of Skydiver 2'
where ' represents after separation.
The momentum of the first skydiver in the x and y directions immediately after separation would be zero since their velocity components in these directions are zero. Therefore, we only need to find the momentum of the second skydiver in the x and y directions.
Momentum of Skydiver 2' = mass of Skydiver 2 * velocity of Skydiver 2'
Let's assume the x and y components of the velocity of the second skydiver immediately after separation are v2x and v2y, respectively. Then:
Momentum of Skydiver 2' = 57.7 kg * (v2x * i + v2y * j)
Setting the total momentum before separation equal to the total momentum after separation, we have:
Momentum of Skydiver 1 + Momentum of Skydiver 2 = Momentum of Skydiver 1' + Momentum of Skydiver 2'
89.3 kg * (4.43 m/s * i + 4.75 m/s * j + 60.7 m/s * k) + 57.7 kg * (v2x * i + v2y * j) = 57.7 kg * (v2x * i + v2y * j)
From this equation, we can equate the coefficients of the i and j vectors to find the x and y components of the velocity of the second skydiver immediately after separation.
Equating the coefficients of the i vector:
89.3 kg * 4.43 m/s + 57.7 kg * v2x = 57.7 kg * v2x
We can solve this equation to find v2x.
Equating the coefficients of the j vector:
89.3 kg * 4.75 m/s + 57.7 kg * v2y = 57.7 kg * v2y
We can solve this equation to find v2y.
Once we have the values for v2x and v2y, we can calculate the change in kinetic energy of the system using the formula:
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Since the only change in the system is the change in the kinetic energy of the second skydiver, we can calculate it as:
Change in Kinetic Energy = (1/2) * mass of Skydiver 2 * (v2x^2 + v2y^2)
Solving these equations will give us the x- and y-components of the velocity of the second skydiver and the change in kinetic energy of the system.