A ball of mass 0.075 kg is fired horizontally into a ballistic pendulum. The pendulum mass is 0.350 kg. The ball is caught in the pendulum, and the centerof mass of the system rises a vertical distance of 0.145 m in the earth's gravitational field. What was the original speed of the ball? Assume g=9.80 m/s^2. How much kinetic energy was lost in the collision?
The distance than the block rises with the ball inside will tell you the velocity right after impact (V2), using conservation of energy.
V2 = sqrt (2gH); H = 0.145 m
Once you have determined V2, get the original speed of the ball (V1) by using conservation of momentum.
m V1 = (M+m) V2
Once you know v and V, calculatring the loss of kinetic energy is easy.
KE loss = (1/2)m V1^2 - (1/2)(m+M)V2^2
9.55
Why did the ball join the circus?
Because it wanted to become a pendulum-ist!
To calculate the original speed of the ball, we can use the principle of conservation of momentum. Since the total momentum before the collision is equal to the total momentum after the collision in an isolated system, we have:
(initial mass of ball) * (initial velocity of ball) = (final mass of ball + pendulum) * (combined velocity after collision)
Using this equation, we can solve for the initial velocity of the ball:
(0.075 kg) * (initial velocity of ball) = (0.075 kg + 0.350 kg) * (velocity after collision)
Simplifying this equation, we get:
(0.075 kg) * (initial velocity of ball) = (0.425 kg) * (velocity after collision)
Now, let's calculate the velocity after collision using the information given. The center of mass rises a vertical distance of 0.145 m, which means the gravitational potential energy gained by the system is:
Potential Energy = (mass of system) * (acceleration due to gravity) * (vertical distance)
Potential Energy = (0.075 kg + 0.350 kg) * (9.8 m/s^2) * (0.145 m)
Now, we know that this gain in potential energy is equal to the loss in kinetic energy. So:
Loss in Kinetic Energy = Potential Energy
Substituting the values, we can solve for the loss in kinetic energy.
Now, if only the ball had learned juggling, it could've balanced its energy better!
To find the original speed of the ball, we can use the principle of conservation of momentum. In an isolated system, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Since the ball is fired horizontally, it has no vertical velocity initially.
Let's define the following variables:
m1 = mass of the ball, which is 0.075 kg
m2 = mass of the pendulum, which is 0.350 kg
v1 = velocity of the ball before collision
v2 = velocity of the pendulum after collision
Since the ball is caught by the pendulum and the two objects move together, we can consider them as a single system.
Before the collision, the momentum is only contributed by the ball:
momentum_before = m1 * v1
After the collision, the momentum is contributed by the ball and the pendulum:
momentum_after = (m1 + m2) * v2
Since momentum is conserved, we can equate the two expressions:
m1 * v1 = (m1 + m2) * v2
Rearranging the equation:
v1 = (m1 + m2) * v2 / m1
Substituting the given values:
v1 = (0.075 kg + 0.350 kg) * v2 / 0.075 kg
Now, let's find the value of v2.
When the ball is caught by the pendulum, the center of mass of the system rises by a vertical distance of 0.145 m. This change in height is due to the gravitational potential energy gained by the system.
The change in gravitational potential energy can be calculated using the formula:
ΔPE = mgh
Where:
ΔPE = change in potential energy
m = mass of the system
g = acceleration due to gravity (9.8 m/s^2)
h = change in height (0.145 m)
Since the ball and pendulum move together, the mass of the system is equal to the combined mass of the ball and the pendulum:
m = m1 + m2 = 0.075 kg + 0.350 kg = 0.425 kg
Substituting the values:
ΔPE = 0.425 kg * 9.8 m/s^2 * 0.145 m
Now, let's find the change in kinetic energy of the system.
The initial kinetic energy of the system is given by:
KE_initial = (1/2) * m1 * v1^2
The final kinetic energy of the system is given by:
KE_final = (1/2) * (m1 + m2) * v2^2
The change in kinetic energy is given by:
ΔKE = KE_initial - KE_final
Substitute the known values into the equations and solve for v1 and ΔKE:
v1 = (0.075 kg + 0.350 kg) * v2 / 0.075 kg
ΔPE = 0.425 kg * 9.8 m/s^2 * 0.145 m
KE_initial = (1/2) * m1 * v1^2
KE_final = (1/2) * (m1 + m2) * v2^2
ΔKE = KE_initial - KE_final
Calculate these values to find the original speed of the ball and the change in kinetic energy.