Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 34

Chapter 11, Problem 34

A 20 g ball of clay traveling east at 3.0 m/s collides with a 30 g ball of clay traveling north at 2.0 m/s. What are the speed and the direction of the resulting 50 g ball of clay? Give your answer as an angle north of east.

Verified step by step guidance

Step 1: Identify the type of collision. This is an inelastic collision because the two balls of clay stick together after the collision, forming a single mass. The law of conservation of momentum applies here.

Step 2: Write the conservation of momentum equations for the x-direction (east-west) and y-direction (north-south). The total momentum before the collision equals the total momentum after the collision. For the x-direction: \( p_x = m_1 v_{1x} + m_2 v_{2x} \). For the y-direction: \( p_y = m_1 v_{1y} + m_2 v_{2y} \).

Step 3: Substitute the given values into the momentum equations. For the x-direction, \( m_1 = 0.020 \; \text{kg}, v_{1x} = 3.0 \; \text{m/s}, m_2 = 0.030 \; \text{kg}, v_{2x} = 0 \; \text{m/s} \) (since the second ball has no velocity in the x-direction). For the y-direction, \( v_{1y} = 0 \; \text{m/s} \) (since the first ball has no velocity in the y-direction), and \( v_{2y} = 2.0 \; \text{m/s} \).

Step 4: Calculate the total momentum in the x-direction and y-direction. Use the equations \( p_x = m_1 v_{1x} + m_2 v_{2x} \) and \( p_y = m_1 v_{1y} + m_2 v_{2y} \). Then, find the magnitude of the total momentum \( p_{\text{total}} \) using \( p_{\text{total}} = \sqrt{p_x^2 + p_y^2} \).

Step 5: Determine the speed and direction of the resulting ball of clay. The speed is given by \( v = \frac{p_{\text{total}}}{m_{\text{total}}} \), where \( m_{\text{total}} = 0.050 \; \text{kg} \). The direction (angle north of east) is found using \( \theta = \arctan\left(\frac{p_y}{p_x}\right) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In collisions, the momentum before the collision equals the momentum after the collision. This concept is crucial for solving problems involving collisions, as it allows us to calculate the resulting velocities of objects post-collision.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this scenario, the velocities of the two clay balls must be treated as vectors, taking into account their magnitudes and directions. The resultant velocity can be found using the Pythagorean theorem and trigonometric functions to determine both the magnitude and direction of the combined velocity.

Resultant Velocity and Direction

The resultant velocity is the vector sum of the individual velocities of the colliding objects. To find the direction of the resultant velocity, one must calculate the angle it makes with a reference direction, typically using trigonometric functions like tangent. This angle indicates the direction of the new object formed after the collision, which is essential for providing a complete answer to the problem.

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