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 33b

Chapter 11, Problem 33b

At the center of a 50-m-diameter circular ice rink, a 75 kg skater traveling north at 2.5 m/s collides with and holds on to a 60 kg skater who had been heading west at 3.5 m/s. Where will they reach it? Give your answer as an angle north of west.

Verified step by step guidance

Step 1: Recognize that this is a problem involving conservation of momentum. Since the skaters collide and stick together, it is an example of an inelastic collision. Momentum is conserved in both the north-south and east-west directions.

Step 2: Write the equation for conservation of momentum in the north-south direction. The momentum before the collision is given by the mass of the first skater multiplied by their velocity. After the collision, the combined mass of the two skaters moves with a final velocity in the north-south direction. Use the formula:

p_{y} = m_{1} v_{1}

Step 3: Write the equation for conservation of momentum in the east-west direction. The momentum before the collision is given by the mass of the second skater multiplied by their velocity. After the collision, the combined mass of the two skaters moves with a final velocity in the east-west direction. Use the formula:

p_{x} = m_{2} v_{2}

Step 4: Combine the momentum components to find the resultant velocity vector. Use the Pythagorean theorem to calculate the magnitude of the final velocity:

v = \sqrt{{v_{x}}^{2} + {v_{y}}^{2}}

Step 5: Determine the direction of the final velocity vector. Use the tangent function to calculate the angle north of west:

θ = \tan (\frac{v_{y}}{v_{x}})

. This angle gives the direction of the combined skaters' motion relative to the west direction.

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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 conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, the skaters collide and hold onto each other, meaning we can calculate their combined momentum in both the north-south and east-west directions to find their final velocity after the collision.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this case, the velocities of the two skaters must be treated as vectors, taking into account their directions. The resultant vector will help determine the final direction and speed of the combined skaters after the collision.

Angle Measurement in Navigation

In navigation, angles are often measured relative to cardinal directions, such as north and west. To express the final direction of the skaters after the collision, we need to calculate the angle north of west, which involves using trigonometric functions to find the angle based on the resultant vector's components.

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