A 3.20 kg mud ball has a perfectly inelastic collision with a second mud ball that is initially at rest. The composite system moves with a speed equal to one-half the original speed of the 3.20 mud ball. What is the mass of the second mud ball?
kg
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
The law of conservation of momentum states that the total momentum of a system before and after a collision remains the same, provided no external forces act on the system.
In this case, we have a perfectly inelastic collision. When two objects stick together after colliding, their masses are combined into a single composite object.
Let's denote the mass of the second mud ball as m2.
The initial momentum of the system can be calculated as:
Initial momentum = mass of the first mud ball * initial velocity of the first mud ball
The final momentum of the system can be calculated as:
Final momentum = Total mass of the composite object after collision * final velocity of the composite object
Since the final velocity of the composite object is one-half the initial velocity of the first mud ball, we can write it as 1/2 * initial velocity.
Using the conservation of momentum, we can equate the initial and final momenta:
mass of the first mud ball * initial velocity of the first mud ball = Total mass of the composite object after collision * 1/2 * initial velocity
2. Conservation of kinetic energy:
In a perfectly inelastic collision, kinetic energy is not conserved. Therefore, we cannot directly use the conservation of kinetic energy to solve this problem.
Now, let's substitute the known values into the momentum equation and solve for the unknown mass of the second mud ball:
mass of the first mud ball * initial velocity of the first mud ball = (mass of the first mud ball + mass of the second mud ball) * 1/2 * initial velocity
Substituting the given values: mass of the first mud ball = 3.20 kg, initial velocity of the first mud ball = v, and final velocity of the composite object = 1/2 * v, we get:
(3.20 kg) * v = (3.20 kg + mass of the second mud ball) * (1/2 * v)
Simplifying the equation, we find:
3.20 kg * v = 1.60 kg * v + (mass of the second mud ball) * (1/2 * v)
Subtracting 1.60 kg * v from both sides:
1.60 kg * v = (mass of the second mud ball) * (1/2 * v)
Now, we can cancel out v from both sides:
1.60 kg = (mass of the second mud ball) * (1/2)
Finally, solving for the mass of the second mud ball:
(mass of the second mud ball) = (1.60 kg) * (2)
mass of the second mud ball = 3.20 kg
Hence, the mass of the second mud ball is 3.20 kg.
To find the mass of the second mud ball, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the mass of the second mud ball as m2.
Before the collision:
Momentum of the first mud ball (m1) = m1 * v1 (original speed of the 3.20 kg mud ball)
After the collision:
Composite system momentum = (m1 + m2) * (1/2 * v1) (speed equal to one-half the original speed of the 3.20 mud ball)
According to the conservation of momentum, these two values should be equal.
m1 * v1 = (m1 + m2) * (1/2 * v1)
Now let's solve for m2:
m1 * v1 = (m1 + m2) * (1/2 * v1)
Expanding the equation:
m1 * v1 = (1/2) * (m1 * v1 + m2 * v1)
Multiply both sides by 2 to cancel out the 1/2:
2 * m1 * v1 = m1 * v1 + m2 * v1
Simplifying:
2 * m1 * v1 - m1 * v1 = m2 * v1
m1 * v1 = m2 * v1
Now we can cancel out v1 from both sides:
m1 = m2
Therefore, the mass of the second mud ball (m2) is equal to the mass of the first mud ball (m1), which is 3.20 kg.