He drops a ninja ball device from a height of 10.0m. The mass of the enter device is 190g and the mass of the top (smallest) ball is 2.90g
A) calculate the momentum of the device as it hits the ground (ignore air resistance) I know that this answer is 2.66 for sure
B) if the entire device comes to rest, expect for the top ball, what will the speed of the top ball be immediately after hitting the ground?
C) how high will the ball rise?
D) ignore air resistance, from what height would the you have to drop the device so that after hitting the ground, the top ball achieves escape velocity and leaving the earth entirely?! Recall the escape velocity from earth=11.2km/s
I know it's kinda a lot but I really need help can you also please show all work thank u a lot
A) To calculate the momentum of the device as it hits the ground, we will use the formula for momentum:
Momentum = mass * velocity
In this case, the mass of the entire device is 190g, which is 0.19kg. The velocity is the unknown we need to calculate.
We can use the principle of conservation of energy to calculate the velocity just before the device hits the ground. We can equate the initial potential energy of the device with the final kinetic energy:
Initial potential energy = Final kinetic energy
The initial potential energy is given by:
Initial potential energy = m * g * h
where m is the mass of the entire device (0.19kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the device is dropped (10.0m).
The final kinetic energy is given by:
Final kinetic energy = (1/2) * m * v²
where m is the mass of the entire device (0.19kg) and v is the velocity just before hitting the ground.
Setting the initial potential energy equal to the final kinetic energy:
m * g * h = (1/2) * m * v²
Simplifying the equation by canceling out the mass:
g * h = (1/2) * v²
Now we can solve for v:
v² = 2 * g * h
v = sqrt(2 * g * h)
Substituting the given values for g (approximately 9.8 m/s²) and h (10.0m):
v = sqrt(2 * 9.8 * 10.0) ≈ 14.0 m/s
Finally, we can calculate the momentum using the formula:
Momentum = mass * velocity
Momentum = 0.19kg * 14.0 m/s
Momentum ≈ 2.66 kg.m/s
So, the momentum of the device as it hits the ground is approximately 2.66 kg.m/s.
B) If the entire device comes to rest, except for the top ball, we can find the speed of the top ball immediately after hitting the ground using the principle of conservation of momentum. The momentum before the collision should be equal to the momentum after the collision (while ignoring air resistance):
Initial momentum = Final momentum
The initial momentum is given by the mass of the entire device multiplied by its velocity just before hitting the ground:
Initial momentum = mass * velocity
Initial momentum = 0.19kg * 14.0 m/s
The final momentum is given by the mass of the top ball multiplied by its speed immediately after hitting the ground. Let's call the speed of the top ball v_top.
Final momentum = mass_top * v_top
Since the entire device comes to rest except for the top ball, the mass of the top ball is the same as the mass of the top (smallest) ball, which is 2.90g, or 0.00290kg.
Setting the initial momentum equal to the final momentum:
0.19kg * 14.0 m/s = 0.00290kg * v_top
Simplifying the equation:
v_top = (0.19kg * 14.0 m/s) / 0.00290kg
v_top ≈ 922.07 m/s
So, the speed of the top ball immediately after hitting the ground is approximately 922.07 m/s.
C) To calculate how high the ball will rise after hitting the ground, we can use the principle of conservation of mechanical energy. The potential energy before the ball hits the ground will be converted into kinetic energy as it rises.
The initial potential energy is given by:
Initial potential energy = mass_top * g * h
where mass_top is the mass of the top ball (0.00290kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height the ball will rise.
The final kinetic energy is given by:
Final kinetic energy = (1/2) * mass_top * v_top²
where mass_top is the mass of the top ball (0.00290kg) and v_top is the speed of the top ball immediately after hitting the ground (922.07 m/s).
Setting the initial potential energy equal to the final kinetic energy:
mass_top * g * h = (1/2) * mass_top * v_top²
Simplifying the equation by canceling out the mass_top:
g * h = (1/2) * v_top²
h = (1/2) * v_top² / g
Substituting the given values:
h = (1/2) * (922.07 m/s)² / 9.8 m/s²
h ≈ 43,689.10 m
So, the ball will rise to a height of approximately 43,689.10 meters.
D) To find the height from which you would have to drop the device so that after hitting the ground, the top ball achieves escape velocity and leaves the Earth entirely, we first need to find the velocity the top ball needs to reach. Escape velocity from Earth is approximately 11.2 km/s, which is 11,200 m/s.
Using the principle of conservation of energy, we can again equate the initial potential energy with the final kinetic energy:
Initial potential energy = Final kinetic energy
The initial potential energy is given by:
Initial potential energy = mass_top * g * h
where mass_top is the mass of the top ball (0.00290kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the device is dropped (unknown).
The final kinetic energy is given by:
Final kinetic energy = (1/2) * mass_top * v_escape²
where mass_top is the mass of the top ball (0.00290kg) and v_escape is the escape velocity (11,200 m/s).
Setting the initial potential energy equal to the final kinetic energy:
mass_top * g * h = (1/2) * mass_top * v_escape²
Simplifying the equation by canceling out the mass_top:
g * h = (1/2) * v_escape²
h = (1/2) * v_escape² / g
Substituting the given values:
h = (1/2) * (11,200 m/s)² / 9.8 m/s²
h ≈ 6,140,816.33 m
So, you would have to drop the device from a height of approximately 6,140,816.33 meters (or about 6,140.8 km) to achieve escape velocity and have the top ball leave the Earth entirely.