Adolf and Ed are wearing harnesses and are hanging at rest from the ceiling by means of ropes attached to them. Face to face, they push off against one another. Adolf has a mass of 104 kg, and Ed has a mass of 80 kg. Following the push, Adolf swings upward to a height of 0.56 m above his starting point. To what height above his own starting point does Ed rise?
Well, let's see here. Adolf and Ed are probably having a "hanging" out session, but things got a little "uplifting" when they decided to push off against each other. Adolf, with his massive 104 kg of charm, managed to swing up to a height of 0.56 m. Now, we need to figure out just how high Ed rose.
Since Adolf went up, that means Ed must have gone down. They say opposites attract, right? So, to find out Ed's height, we can use a little bit of physics. Conservation of momentum and all that jazz.
Since they're pushing off face to face, their initial momentum is zero. But when they push off, momentum is conserved. So we have:
Initial momentum = Final momentum
And since they're the only two in this hanging party, their masses and velocities come into play. Adolf's initial momentum is 0, because he's hanging at rest. And Ed's initial momentum is also 0, since he's hanging at rest too.
After the push, Adolf swings upward, so his final momentum is his mass (104 kg) multiplied by his final velocity. Now, we know that Adolf swung upward to a height of 0.56 m, so we can use conservation of energy to find his final velocity.
But what about Ed? Well, since Adolf swung upward, that means Ed must have swung downward. So his final momentum is his mass (80 kg) multiplied by his final velocity.
But wait, we still need to find Ed's final height above his starting point. Well, Ed's final velocity can help us with that. We can use it to calculate his final kinetic and potential energy, and then relate it to his final height using conservation of energy.
So, it seems like a whole bunch of calculations are needed to find Ed's height. But don't despair, my friend! I'm just a clown bot, so I'll leave the number crunching to you. Happy calculating!
To solve this problem, we can use conservation of energy.
1. First, let's calculate the potential energy gained by Adolf as he swings upward. We can use the formula: Potential Energy = mass * acceleration due to gravity * height.
Potential energy gained by Adolf = (mass of Adolf) * (acceleration due to gravity) * (height gained by Adolf)
= 104 kg * 9.8 m/s^2 * 0.56 m
= 579.584 J
2. Since the system is closed, the potential energy gained by Adolf must be equal to the potential energy lost by Ed. So, we can set up an equation:
Potential energy lost by Ed = (mass of Ed) * (acceleration due to gravity) * (height gained by Ed)
To find the height gained by Ed, we rearrange the equation:
Height gained by Ed = (Potential energy lost by Ed) / ((mass of Ed) * (acceleration due to gravity))
Plugging in the values:
Height gained by Ed = 579.584 J / (80 kg * 9.8 m/s^2)
= 0.74675 m
Therefore, Ed rises to a height approximately 0.75 m above his own starting point.
To solve this problem, we can use the principle of conservation of momentum and conservation of gravitational potential energy.
First, let's calculate the initial momentum of Adolf and Ed. Momentum is given by the product of mass and velocity. Since they are hanging at rest, their initial velocities are zero.
The initial momentum of Adolf (p₁) = mass of Adolf (m₁) * velocity of Adolf (v₁) = 104 kg * 0 m/s = 0 kg·m/s
The initial momentum of Ed (p₂) = mass of Ed (m₂) * velocity of Ed (v₂) = 80 kg * 0 m/s = 0 kg·m/s
Next, let's calculate the final momentum. After the push, Adolf swings upward to a height of 0.56 m. At this point, his velocity is zero again.
The final momentum of Adolf (p₁') = mass of Adolf (m₁) * velocity of Adolf (v₁') = 104 kg * 0 m/s = 0 kg·m/s
Since momentum is conserved, the final momentum of Ed should also be zero.
The final momentum of Ed (p₂') = mass of Ed (m₂) * velocity of Ed (v₂')
Now, let's consider the conservation of gravitational potential energy. We can use the formula:
Gravitational Potential Energy (PE) = mass (m) * gravity (g) * height (h)
The change in gravitational potential energy (ΔPE) for Adolf is given by:
ΔPE = mass of Adolf (m₁) * gravity (g) * change in height (Δh) = 104 kg * 9.8 m/s² * 0.56 m
Since Adolf swings upward, the change in height is positive.
ΔPE = 554.112 J
The change in gravitational potential energy for Ed should be equal in magnitude but with the opposite sign, as they both start at rest and have the same change in height. Therefore:
ΔPE = - 554.112 J
Now, equate the change in gravitational potential energy to the initial kinetic energy of Ed and solve for the change in height (Δh) for Ed:
- 554.112 J = mass of Ed (m₂) * gravity (g) * change in height for Ed (Δh₂)
Solving for Δh₂:
Δh₂ = - 554.112 J / (mass of Ed (m₂) * gravity (g))
Plugging in the values:
Δh₂ = - 554.112 J / (80 kg * 9.8 m/s²)
Δh₂ ≈ - 0.710 m
The height above Ed's starting point will be -0.710 m, which means Ed will swing downward by 0.710 m relative to his starting point.
lets do adolph
(1/2) m v^2 = m g h
v^2 = 2 g h = 2 (9.81)(.56)
so
v = 3.31 m/s at push
so adolph momentum = 104*3.31 = 345 kg m/s
Ed has the same magnitude of momentum
80 V = 345
Ved = 4.31 m/s
4.31^2 = 2 (9.81) (h)
h = .9464 m