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Chapter 1: Problem 1

An object \(A\) moving with velocity \(v\) collides with a stationary object \(B\). After the collision, \(A\) is moving with velocity \(\frac{1}{2} \boldsymbol{v}\) and \(B\) with velocity \(\frac{3}{2} \boldsymbol{v}\). Find the ratio of their masses. If, instead of bouncing apart, the two bodies stuck together after the collision, with what velocity would they then move?

Short Answer

Expert verified
Answer: The ratio of the masses of objects A and B after the collision is m_B/m_A = 1/3, and the velocity of both objects after they stick together in the collision is (3/4)v.

Step by step solution

01

Understand conservation of momentum

Conservation of momentum states that the total momentum of a closed system remains constant before and after a collision. In this case, A and B are our objects in the system.
02

Find initial and final momentum

Initially, object A has a velocity v, and object B is not moving. Let the mass of object A be m_A and the mass of object B be m_B. The initial momentum of the system is given by the momentum of object A: P_initial = m_A * v. Since object B is stationary, its momentum is zero. After the collision, object A is moving with velocity (1/2)v and object B is moving with velocity (3/2)v. The total momentum after the collision is: P_final = m_A * (1/2)v + m_B * (3/2)v.
03

Use conservation of momentum

According to conservation of momentum, P_initial = P_final. Thus, we have: m_A * v = m_A * (1/2)v + m_B * (3/2)v.
04

Find the ratio of masses

Now, solve for the ratio m_B/m_A: Divide the equation by v to isolate the masses: m_A = (1/2)m_A + (3/2)m_B. Subtract (1/2)m_A from both sides of the equation: (1/2)m_A = (3/2)m_B. Now, divide both sides by (1/2)m_A: 1 = 3(m_B/m_A). Thus, we get the ratio of the masses: m_B/m_A = 1/3.
05

Find the velocity of the combined objects

If the two objects stick together after the collision, their combined mass is (m_A + m_B). Since momentum is conserved, the combined velocity (V') of the objects can be calculated using the initial total momentum (P_initial = m_A * v): m_A*v = (m_A + m_B) * V'. Substitute m_B = (1/3)m_A: m_A*v = (m_A + (1/3)m_A) * V'. Combine the masses and divide by m_A: v = (4/3) * V'. Finally, find the combined velocity V': V' = (3/4)v. The combined object would move with a velocity of (3/4)v after the collision if they were to stick together.

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