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Chapter 8: Problem 20

You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a 0.600-kg ball that is traveling horizontally at 10.0 m/s. Your mass is 70.0 kg. (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so afterward it is moving horizontally at 8.0 m/s in the opposite direction, what is your speed after the collision?

Short Answer

Expert verified
(a) 0.085 m/s; (b) 0.154 m/s.

Step by step solution

01

Understand the Conservation of Momentum

In a closed and isolated system, the total linear momentum of an object remains constant if no external forces act upon it. The equation for momentum conservation is: \( m_1 \cdot v_1 + m_2 \cdot v_2 = (m_1 + m_2) \cdot V'\), where \(m_1\) and \(v_1\) are the mass and velocity of the first object, \(m_2\) and \(v_2\) are the mass and velocity of the second object, and \(V'\) is the final velocity of the system.
02

Apply Momentum Conservation for Catching the Ball

When you catch the ball, we consider both you and the ball as the system. The initial momentum of the ball is \(0.600 \times 10.0 = 6.0 \, \text{kg m/s}\). You are initially at rest, so your momentum is \(70.0 \times 0 = 0\). The total initial momentum is therefore \(6.0 \, \text{kg m/s}\). After catching the ball, the total mass is \(70.6 \, \text{kg}\). Thus, \(6.0 = 70.6 \times V'\). Solve for \(V'\) to find the velocity.
03

Calculate Final Velocity After Catching the Ball

To find \(V'\), solve the equation: \(V' = \frac{6.0}{70.6} \\approx 0.085 \, \text{m/s}\). Therefore, the velocity of you and the ball together after catching is 0.085 m/s.
04

Analyze the Ball Bouncing Off Scenario

In the second scenario, the ball bounces back at 8.0 m/s, so the ball's momentum after collision is \(0.600 \times (-8.0) = -4.8 \, \text{kg m/s}\). Initially, the ball's momentum was 6 kg m/s. Thus, using conservation of momentum: \(6.0 = 70 \cdot V' + (-4.8)\). Solve for \(V'\).
05

Calculate Final Velocity After Ball Bounces Off

Rearrange the equation from Step 4: \(6.0 + 4.8 = 70 \cdot V'\) which simplifies to \(10.8 = 70 \cdot V'\). Solving for \(V'\), we find \(V' = \frac{10.8}{70} \approx 0.154 \, \text{m/s}\). Thus, your speed after being hit by the ball is 0.154 m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum
Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is determined by both the object's mass and its velocity, given by the formula: \[ p = m \cdot v \] where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. The unit of momentum is typically kilogram meter per second (kg m/s).
- Momentum is a vector quantity, meaning it has both magnitude and direction.
- In a closed system, where no external forces are acting, the total momentum before any event is equal to the total momentum after the event, known as the Conservation of Momentum.
In the given problem, when you catch the ball on ice, the total momentum before and after catching the ball remains constant. This conservation principle helps us understand how velocities change due to the interactions.
Elastic Collision
An elastic collision is a type of collision where both momentum and kinetic energy are conserved. These collisions occur when objects collide and then move apart without deformation or generation of heat.
- The total kinetic energy before the collision equals the total kinetic energy after the collision.
- The formulae involved ensure both the mathematical conservation of momentum and kinetic energy. In the exercise scenario, when the ball bounces off your chest, the event is not perfectly elastic due to the kinetic energy difference, but it introduces the principle. Elastic collisions, though not perfectly replicable in most real-world scenarios, provide a simplified framework for initial understanding.
Inelastic Collision
An inelastic collision is another collision type where momentum is conserved but kinetic energy is not necessarily conserved. These are more common in everyday interactions, where some kinetic energy is converted to other forms like heat or sound.
- Often, the colliding objects stick together post-collision, moving as a single mass.
- Unlike elastic collisions, the total kinetic energy post-collision can be less than prior. For the given problem, when you catch the ball and move together with it, this is an example of a perfectly inelastic collision. The ball and you move together with a new shared velocity. This practical situation helps illustrate how energy conversions differ from those in elastic collisions, underscoring how real-world events might translate theory into practice.

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Most popular questions from this chapter

To warm up for a match, a tennis player hits the 57.0-g ball vertically with her racket. If the ball is stationary just before it is hit and goes 5.50 m high, what impulse did she impart to it?

You are standing on a concrete slab that in turn is resting on a frozen lake. Assume there is no friction between the slab and the ice. The slab has a weight five times your weight. If you begin walking forward at 2.00 m/s relative to the ice, with what speed, relative to the ice, does the slab move?

You are standing on a large sheet of frictionless ice and holding a large rock. In order to get off the ice, you throw the rock so it has velocity 12.0 m/s relative to the earth at an angle of 35.0\(^\circ\) above the horizontal. If your mass is 70.0 kg and the rock's mass is 3.00 kg, what is your speed after you throw the rock? (See Discussion Question Q8.7.)

One 110-kg football lineman is running to the right at 2.75 m/s while another 125-kg lineman is running directly toward him at 2.60 m/s. What are (a) the magnitude and direction of the net momentum of these two athletes, and (b) their total kinetic energy?

You are called as an expert witness to analyze the following auto accident: Car \(B\), of mass 1900 kg, was stopped at a red light when it was hit from behind by car \(A\), of mass 1500 kg. The cars locked bumpers during the collision and slid to a stop with brakes locked on all wheels. Measurements of the skid marks left by the tires showed them to be 7.15 m long. The coefficient of kinetic friction between the tires and the road was 0.65. (a) What was the speed of car \(A\) just before the collision? (b) If the speed limit was 35 mph, was car \(A\) speeding, and if so, by how many miles per hour was it \(exceeding\) the speed limit?
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