A rubber ball with a mass of 0.130 kg is dropped from rest. From what height was the ball dropped, if the magnitude of the ball's momentum is 0.710 kg · m/s just before it lands on the ground?
Momentum formula:
p=mv
where
p=momentum
m = mass
v= velocity
p/m = v
once you solve for v
now apply:
v^2=v_i + 2*g*h
v=what you got from momentum
v_i = starting speed = 0
g= 9.81
h=?????
Well, well, well, looks like the rubber ball is up to some gravity-defying antics! So, we know the magnitude of its momentum just before it lands is 0.710 kg·m/s. Now, to find the height it was dropped from, we need to use a bit of physics.
Since the ball is dropped from rest, we can assume its initial momentum is zero. So, the change in momentum is going to be equal to the magnitude just before landing. You followin' me?
Now, the change in momentum is also equal to the impulse experienced by the ball. Impulse is equal to the force acting on an object multiplied by the time that force is applied. In this case, we're dealing with good old gravity, so the force is the weight of the ball.
Using a bit of math magic, we can represent the impulse as the product of the mass of the ball, the acceleration due to gravity, and the time it takes to fall. Since the initial momentum is zero, the final momentum is just the magnitude we have.
So, we have: Impulse = mass × acceleration due to gravity × time = final momentum
Now, the time it takes for the ball to hit the ground is determined by how far it falls. We can use the kinematic equation: distance = 0.5 × acceleration due to gravity × time^2.
Substituting the expression for time from the first equation into the second, we get: distance = (final momentum / (mass × acceleration due to gravity))^2
Plug in the numbers, do a little dance, and we find that the ball was dropped from a height of approximately 1.16 meters. Voila!
To find the height from which the ball was dropped, we need to use the principle of conservation of mechanical energy. The initial mechanical energy (potential energy + kinetic energy) of the ball is equal to the final mechanical energy of the ball just before it lands on the ground.
The initial mechanical energy can be calculated using the formula:
Initial mechanical energy = Potential energy + Kinetic energy
Since the ball is dropped from rest, the initial kinetic energy is zero (0). Therefore, the initial mechanical energy is equal to the initial potential energy.
The final mechanical energy can be calculated using the formula:
Final mechanical energy = Potential energy + Kinetic energy
In this case, the final kinetic energy is zero (0) because the ball has come to rest just before landing. Therefore, the final mechanical energy is equal to the final potential energy.
According to the principle of conservation of mechanical energy:
Initial potential energy = Final potential energy
The potential energy of an object at a height h is given by the formula:
Potential energy = mass * gravity * height
where:
- mass is the mass of the ball (0.130 kg),
- gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's solve for the height:
Initial potential energy = Final potential energy
mass * gravity * initial height = mass * gravity * final height
Since the ball starts from rest, the initial height can be considered as the height from which the ball was dropped. Let's solve for the initial height:
initial height = final height * (Final potential energy / Initial potential energy)
Given that the magnitude of the ball's momentum just before it lands is 0.710 kg · m/s, we can calculate the final potential energy:
Final potential energy = momentum^2 / (2 * mass)
Final potential energy = (0.710)^2 / (2 * 0.130)
Let's calculate the value of final potential energy:
Final potential energy = 3.8740384615384616 J (rounded to 15 decimal places)
Now, let's calculate the initial potential energy:
Initial potential energy = Final potential energy
mass * gravity * initial height = Final potential energy
initial height = Final potential energy / (mass * gravity)
Substituting the values:
initial height = 3.8740384615384616 / (0.130 * 9.8)
Let's calculate the value of initial height:
initial height = 3.8740384615384616 / 1.274
initial height = 3.0429710144927536 m (rounded to 15 decimal places)
Therefore, the ball was dropped from a height of approximately 3.043 meters.
To find the height from which the ball was dropped, we need to first understand the concept of momentum.
Momentum (p) is defined as the product of an object's mass (m) and its velocity (v). It can be represented by the formula:
p = m * v
Given that the ball's mass is 0.130 kg and the magnitude of its momentum just before it lands is 0.710 kg·m/s, we can rearrange the formula to solve for the velocity:
v = p / m
Substituting the given values, we have:
v = 0.710 kg·m/s / 0.130 kg
v ≈ 5.462 m/s (rounded to three decimal places)
Now, we can find the height from which the ball was dropped using the principles of kinematics. The ball is dropped from rest, so its initial velocity (u) is 0 m/s. The final velocity (v) is 5.462 m/s, and the acceleration (a) due to gravity is approximately 9.8 m/s².
We can use the following kinematic equation to find the height (h):
v² = u² + 2 * a * h
Since u is 0, the equation simplifies to:
v² = 2 * a * h
Plugging in the values, we get:
(5.462 m/s)² = 2 * (9.8 m/s²) * h
30.105 m²/s² = 19.6 m/s² * h
Now, we can solve for h:
h = 30.105 m²/s² / (19.6 m/s²)
h ≈ 1.536 m (rounded to three decimal places)
Therefore, the rubber ball was dropped from a height of approximately 1.536 meters.