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 117 kg, and Ed has a mass of 77 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
Well, it looks like Adolf had a "Mein Kampf" upwards! Now let's figure out how high Ed rises.
To solve this, we can use the principle of conservation of momentum, which states that the total momentum before the push is equal to the total momentum after the push. Since there are no external forces acting on Adolf and Ed, we can assume that the momentum is conserved.
Given that Adolf swings upward to a height of 0.65 m, we know that he gained potential energy equal to m * g * h, where m is his mass, g is the acceleration due to gravity, and h is the height.
Now, let's apply the principle of conservation of momentum. Initially, both Adolf and Ed are stationary, so the total momentum before the push is zero. After the push, their combined momentum is still zero since they push off against each other.
Since the total momentum is zero, the upward momentum of Adolf must be equal to the downward momentum of Ed.
So, we have:
Mass of Adolf * Velocity of Adolf = Mass of Ed * Velocity of Ed
Now, to find the height Ed rises, we can use the equation for gravitational potential energy:
m * g * h = (Mass of Ed * Velocity of Ed)^2 / (2 * Mass of Ed)
Simplifying this equation will give us the height Ed rises. And just like that, we'll know how high Ed will go.
To solve this problem, we can use the conservation of momentum principle, which states that the total momentum before the push is equal to the total momentum after the push.
Let's assume the initial velocity of Adolf before the push is v1 and the initial velocity of Ed before the push is v2. After the push, their final velocities will be v1' and v2', respectively.
According to the conservation of momentum, we can write the equation:
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2' --(1)
We also know that Adolf rises to a height of 0.65 m above his starting point. As he rises, his potential energy increases while his kinetic energy decreases. The total energy before and after the push remains the same, so we can write:
m1 * g * h1 = m1 * g * h1' + (1/2) * m1 * v1'^2 --(2)
Where,
m1 = mass of Adolf = 117 kg
m2 = mass of Ed = 77 kg
v1 = initial velocity of Adolf
v2 = initial velocity of Ed
v1' = final velocity of Adolf
v2' = final velocity of Ed
g = acceleration due to gravity = 9.8 m/s^2
h1 = starting height = 0 (since they are hanging from the ceiling)
h1' = height attained by Adolf = 0.65 m
We can use equation (2) to find the final velocity of Adolf (v1').
117 * 9.8 * 0 = 117 * 9.8 * 0.65 + (1/2) * 117 * v1'^2
0 = 117 * 9.8 * 0.65 + (1/2) * 117 * v1'^2
v1'^2 = - 2 * 117 * 9.8 * 0.65 / 117
v1'^2 = - 2 * 9.8 * 0.65
v1' ≈ - 7.556 m/s
Now, we can substitute the values of m1, m2, v1, and v1' into equation (1) to find v2':
117 * v1 + 77 * v2 = 117 * v1' + 77 * v2'
117 * (-7.556) + 77 * v2 = 117 * (-7.556) + 77 * v2'
-883.252 + 77 * v2 = -883.252 + 77 * v2'
77 * v2 = 77 * v2'
v2 = v2'
Therefore, the final velocity of Ed (v2') is the same as his initial velocity (v2).
Since Ed is hanging in equilibrium before the push, his initial velocity is 0.
Therefore, v2' = 0 m/s
Now, to find the height Ed rises, we can use equation (2) again:
77 * 9.8 * 0 = 77 * 9.8 * h2 + (1/2) * 77 * 0^2
0 = 77 * 9.8 * h2 + 0
77 * 9.8 * h2 = 0
h2 = 0
Therefore, Ed does not rise above his starting point.
To determine the height above the starting point that Ed rises, we can use the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the push is equal to the total momentum after the push.
The momentum, p, is given by the equation p = mv, where m is the mass and v is the velocity. Since Ed and Adolf are initially at rest, their initial momentum is zero.
After the push, Adolf swings upward to a height of 0.65 m above his starting point. To calculate his final velocity, we can use the conservation of mechanical energy. The change in potential energy, ΔPE, is given by the equation ΔPE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. Since Adolf's final height is 0.65 m, we can calculate his final velocity using the following equation:
ΔPE = mgh = 0.5mv^2
0.5 * 117 * g * 0.65 = 0.5 * 117 * v^2
Simplifying the equation, we find:
v^2 = (g * 0.65)
Now, since momentum is conserved, the total momentum after the push is zero as well. Since Ed has a mass of 77 kg, we can calculate his final velocity using the equation:
0 = mv_adolf + mv_ed
0 = 117 * v_adolf + 77 * v_ed
Rearranging the equation, we find:
v_ed = - (117 / 77) * v_adolf
Substituting the value of v_adolf from the previous calculation, we can solve for v_ed:
v_ed = - (117 / 77) * sqrt(g * 0.65)
Therefore, to determine the height above the starting point that Ed rises to, we can use the equation for potential energy:
ΔPE = mgh_ed
Solving for h_ed, we find:
h_ed = ΔPE / (m * g)
h_ed = (77 * v_ed^2) / (2 * g)
Now, we can substitute the value of v_ed and calculate h_ed.
Please note that the value of the acceleration due to gravity, g, is approximately 9.8 m/s^2.