Adolf and Ed are wearing harnesses and are hanging from the ceiling by means of ropes attached to them. They are face to face and push off against one another. Adolf has a mass of 127 kg, and Ed has a mass of 65 kg. Following the push, Adolf swings upward to a height of 0.65 m above his starting point. To what height above his starting point does Ed rise?
Momentum is conserved.
M1V1=M2V2
Now initial veloicty is proportional to the square root of height attained
(1/2 mv^2=mgh)
so
M1sqrth1=M2sqrtH2
solve for H2.
2.48m
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the push is equal to the total momentum after the push.
The momentum of an object is calculated by multiplying its mass by its velocity. In this case, both Adolf and Ed are initially at rest, so their initial momenta are zero.
After the push, let's assume Adolf moves upward with a velocity v_up and Ed moves downward with a velocity v_down.
Using conservation of momentum, we can write:
(Adolf's mass * v_up) + (Ed's mass * v_down) = 0 (Equation 1)
Now, let's consider the potential energy gained by Adolf. The change in potential energy is given by:
ΔPE = m * g * h
Where ΔPE is the change in potential energy, m is Adolf's mass, g is the acceleration due to gravity, and h is the height above the starting point.
ΔPE_Adolf = m_Adolf * g * h_Adolf
Since Adolf's mass is 127 kg, the acceleration due to gravity is approximately 9.8 m/s², and the height is 0.65 m, we can substitute these values into the equation:
ΔPE_Adolf = 127 kg * 9.8 m/s² * 0.65 m
ΔPE_Adolf = 825.74 J
Since potential energy is converted from kinetic energy, Adolf must have lost 825.74 J of kinetic energy. This means that:
0.5 * m_Adolf * v_up² = 825.74 J (Equation 2)
Similarly, Ed gains the same amount of kinetic energy as Adolf loses. We have:
0.5 * m_Ed * v_down² = 825.74 J (Equation 3)
Now we have three equations:
(1) (m_Adolf * v_up) + (m_Ed * v_down) = 0
(2) 0.5 * m_Adolf * v_up² = 825.74 J
(3) 0.5 * m_Ed * v_down² = 825.74 J
From Equation (1), we can express v_down in terms of v_up:
v_down = - (m_Adolf / m_Ed) * v_up
Substituting this into Equations (2) and (3):
0.5 * m_Adolf * v_up² = 825.74 J
0.5 * m_Ed * (-(m_Adolf / m_Ed) * v_up)² = 825.74 J
Simplifying Equation (3):
(-(m_Adolf / m_Ed) * v_up)² = (2 * 825.74 J) / m_Ed
v_up² = ((2 * 825.74 J) / m_Ed) * (m_Ed / m_Adolf)
Now, we can solve for v_up:
v_up = sqrt(((2 * 825.74 J) / m_Ed) * (m_Ed / m_Adolf))
Substituting the given values:
m_Adolf = 127 kg
m_Ed = 65 kg
v_up = sqrt(((2 * 825.74 J) / 65 kg) * (65 kg / 127 kg))
v_up = sqrt(16.236)
v_up = 4.03 m/s (approximately)
Now we can find v_down:
v_down = - (m_Adolf / m_Ed) * v_up
v_down = - (127 kg / 65 kg) * 4.03 m/s
v_down = -7.87 m/s (approximately)
Finally, we can calculate the height above the starting point for Ed using the same equation for potential energy as before:
ΔPE_Ed = m_Ed * g * h_Ed
h_Ed = ΔPE_Ed / (m_Ed * g)
h_Ed = (825.74 J) / (65 kg * 9.8 m/s²)
h_Ed ≈ 1.26 m
Therefore, Ed rises to a height of approximately 1.26 m above his starting point.
To determine the height to which Ed rises, we can use the law of conservation of momentum, which states that the total momentum before the push is equal to the total momentum after the push in a closed system.
Let's denote Adolf's initial velocity as Va, Ed's initial velocity as Ve, Adolf's final velocity as Vaf, and Ed's final velocity as Vef.
Since Adolf swings upward to a height of 0.65 m above his starting point, we can infer that his initial velocity after the push (Vaf) is zero, as his upward motion comes to a halt at the highest point.
Now, let's consider the conservation of momentum:
Initial momentum = Final momentum
Before the push:
Momentum of Adolf (Ma × Va) + Momentum of Ed (Me × Ve) = (127 kg × Va) + (65 kg × Ve)
After the push:
Momentum of Adolf (Ma × Vaf) + Momentum of Ed (Me × Vef) = 0 (as Vaf = 0)
Therefore, we can rewrite the equation as:
(127 kg × Va) + (65 kg × Ve) = 0
Now, let's solve this equation for Ve:
65 kg × Ve = -127 kg × Va
Ve = (-127 kg × Va) / 65 kg
Since Adolf's mass (127 kg) and initial velocity (Va) are given, we can substitute these values into the equation to calculate Ve:
Ve = (-127 kg × Va) / 65 kg
Since we don't have the values of Adolf's initial velocity, we don't have enough information to determine Ed's height above his starting point.