A 3.49 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-seventh the original speed of the 3.49 mud ball. What is the mass of the second mud ball?
To solve this problem, we can use the law of conservation of momentum.
According to the law of conservation of momentum, 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.
Initially, the first mud ball with a mass of 3.49 kg is moving with a speed v1, and the second mud ball is at rest, so its initial speed is 0.
Therefore, the total momentum before the collision is:
Initial momentum = (mass1 x velocity1) + (mass2 x velocity2)
= (3.49 kg x v1) + (m2 x 0)
= 3.49 kg x v1
After the collision, the total momentum of the composite system is:
Total momentum after collision = (mass1 + mass2) x velocity_final
The problem states that the composite system moves with a speed equal to one-seventh the original speed of the 3.49 mud ball.
So, velocity_final = (1/7) x v1
Therefore, the total momentum after the collision is:
Total momentum after the collision = (3.49 kg + m2) x ((1/7) x v1)
According to the law of conservation of momentum:
Initial momentum = Total momentum after the collision
Hence, we can write the equation as:
3.49 kg x v1 = (3.49 kg + m2) x ((1/7) x v1)
To solve for m2, we can cancel out v1 from both sides of the equation:
3.49 kg = (3.49 kg + m2) x (1/7)
Now, we can solve for m2 by cross multiplying:
3.49 kg x 7 = 3.49 kg + m2
24.43 kg = 3.49 kg + m2
Finally, we can subtract 3.49 kg from both sides of the equation to solve for m2:
m2 = 24.43 kg - 3.49 kg
m2 ≈ 20.94 kg
Therefore, the mass of the second mud ball is approximately 20.94 kg.
To find the mass of the second mud ball, we can use the law of conservation of momentum. The law of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.
The momentum of an object is defined as the product of its mass and velocity. Therefore, the momentum of the first mud ball before the collision can be written as:
Momentum1_before = mass1 * velocity1_before
The momentum of the second mud ball before the collision is zero since it is initially at rest:
Momentum2_before = mass2 * velocity2_before = 0
The total momentum before the collision is the sum of the momentum of the two mud balls:
Total momentum_before = Momentum1_before + Momentum2_before
= mass1 * velocity1_before + 0
= mass1 * velocity1_before
Now, let's consider the momentum after the collision. The composite system moves with a speed equal to one-seventh the original speed of the first mud ball. Therefore, the velocity of the composite system after the collision is (1/7) * velocity1_before.
The momentum of the composite system after the collision can be written as:
Total momentum_after = Mass_composite * velocity_composite_after
Since the composite system consists of the two mud balls, we can write:
Total momentum_after = (mass1 + mass2) * (1/7) * velocity1_before
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore:
mass1 * velocity1_before = (mass1 + mass2) * (1/7) * velocity1_before
Now we can solve the equation for the mass of the second mud ball (mass2):
mass1 * velocity1_before = (mass1 + mass2) * (1/7) * velocity1_before
Dividing both sides of the equation by (1/7) * velocity1_before:
mass1 = (mass1 + mass2)
Subtracting mass1 from both sides of the equation:
mass1 - mass1 = mass2
mass2 = 0
Therefore, the mass of the second mud ball is 0 kg.