Textbook Question
Two balls, of masses mA = 42 g and mB = 96 g, are suspended as shown in Fig. 9–55. The lighter ball is pulled away to a 66° angle with the vertical and released. What is the velocity of the lighter ball before impact?
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11. Momentum & Impulse
Collisions & Motion (Momentum & Energy)
Problem 92
Textbook Question
A rifle is aimed at a 2.0-kg block of wood along an inclined plane making an angle of 25°, as shown in Fig. 9–59. A 9.5-g bullet is fired at 760 m/s and becomes embedded in the block. How far up the incline does the block/bullet slide?
(a) Ignore the friction.
(b) Assume μₖ = 0.33.
Verified step by step guidance1
Step 1: Analyze the problem and identify the key principles. This is a conservation of momentum and energy problem. First, calculate the velocity of the block and bullet system immediately after the collision using the principle of conservation of momentum. The initial momentum of the system is due to the bullet, and the final momentum is shared between the block and the embedded bullet.
Step 2: Use the conservation of momentum formula: \( m_b v_b = (m_b + m_w) v_f \), where \( m_b \) is the mass of the bullet, \( v_b \) is the velocity of the bullet, \( m_w \) is the mass of the block, and \( v_f \) is the final velocity of the block and bullet system. Solve for \( v_f \).
Step 3: For part (a), calculate the distance the block and bullet slide up the incline using the principle of conservation of energy. The initial kinetic energy of the block and bullet system is converted into gravitational potential energy as the system moves up the incline. Use the formula \( KE = PE \), where \( \frac{1}{2}(m_b + m_w)v_f^2 = (m_b + m_w)g h \). Solve for \( h \), the height, and then use trigonometry to find the distance along the incline: \( d = \frac{h}{\sin \theta} \).
Step 4: For part (b), include the effect of friction. The work done against friction reduces the distance the block and bullet slide. The frictional force is given by \( F_f = \mu_k (m_b + m_w) g \cos \theta \). The work done by friction is \( W_f = F_f d \). Modify the energy conservation equation to include the work done by friction: \( \frac{1}{2}(m_b + m_w)v_f^2 = (m_b + m_w)g h + W_f \). Solve for \( d \) iteratively.
Step 5: Summarize the process. For part (a), use conservation of energy without friction to find the distance. For part (b), account for friction by including the work done against friction in the energy equation. Ensure all calculations are consistent with the given values and units.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conservation of Momentum
In a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, when the bullet embeds itself in the block, we can apply the conservation of momentum to find the initial velocity of the block-bullet system immediately after the collision.
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Kinetic Energy and Work-Energy Principle
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. After the bullet embeds in the block, the kinetic energy of the system will be converted into gravitational potential energy as it moves up the incline, allowing us to calculate how far it slides.
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Inclined Plane Dynamics
When analyzing motion on an inclined plane, the forces acting on the object include gravitational force, normal force, and friction (if applicable). The component of gravitational force acting parallel to the incline will affect how far the block slides, especially when considering the angle of the incline and any friction present.
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