Question

Consider these three situations: (i) A ball moving to the right at speed \(v\) is brought to rest. (ii) The same ball at rest is projected at speed \(v\) toward the left. (iii) The same ball moving to the left at speed \(v\) speeds up to \(2 v\). In which situation(s) does the ball undergo the largest change in momentum? a) situation (i) d) situations (i) and (ii) b) situation (ii) e) all three situations c) situation (iii)

Short answer

a) The ball is initially moving to the right with speed v and comes to rest. b) The ball is initially at rest and projected toward the left with speed v. c) The ball is initially moving to the left with speed v and speeds up to 2v. d) Situations (a) and (b) only. e) All three situations. Answer: e) All three situations.

Step-by-step solution

Reviewing the concept of momentum

Momentum is the product of mass and velocity of an object. Mathematically, it is represented as p = m * v, where p is the momentum, m is the mass, and v is the velocity of the object. The change of momentum (∆p) is the difference between the final momentum (pf) and the initial momentum (pi), i.e., ∆p = pf - pi.

Situation (i)

In situation (i), the ball is initially moving to the right at speed v and finally comes to rest. The mass of the ball remains the same, so only the velocity changes. We can find the changes in momentum by calculating: Initial momentum: pi = m * v Final momentum: pf = m * 0 Change in momentum: ∆p = pf - pi = -m * v

Situation (ii)

In situation (ii), the ball is initially at rest and finally projected toward the left at speed v. Again, the mass remains the same, so the changes in momentum are as follows: Initial momentum: pi = m * 0 Final momentum: pf = m * (-v) Change in momentum: ∆p = pf - pi = -m * v

Situation (iii)

In situation (iii), the ball is initially moving to the left at speed v and finally speeds up to 2v. The mass does not change, so we compute the changes in momentum: Initial momentum: pi = m * (-v) Final momentum: pf = m * (-2v) Change in momentum: ∆p = pf - pi = -m * v

Comparing the changes in momentum

Having calculated the changes in momentum for all three situations, we can see that in each situation, the change in momentum is -m * v. Therefore, the ball undergoes the same change in momentum in all three situations. So, the correct answer is: e) all three situations

Key concepts

Momentum

Momentum is a fundamental concept in physics, often described as the 'oomph' an object has due to its motion. It's a vector quantity, meaning it has both magnitude and direction, and is a product of an object's mass (a measure of the amount of matter) and its velocity (the speed and direction of its motion). The mathematical representation of momentum (\textbf{p}) is given by the equation: \[ \mathbf{p} = m \cdot \mathbf{v} \] where \(m\) represents mass and \(\mathbf{v}\) denotes velocity.

Momentum is conserved in a closed system where no external forces are acting. This principle, known as the conservation of momentum, is crucial for understanding collisions and other interactions in physics. Understanding momentum is not just about memorizing the formula, it's about realizing its usefulness in predicting the outcomes of moving objects when they interact.

Change in Momentum Calculation

The change in momentum, which is often referred to as impulse, occurs whenever an object's velocity changes, which can happen due to a change in speed, direction, or both. Calculating this change involves determining the object's momentum at two different instances and then finding the difference between these values.

The formula to calculate the change in momentum (\textbf{∆p}) looks as follows: \[ \mathbf{∆p} = \mathbf{p}_f - \mathbf{p}_i \] Here, \(\mathbf{p}_f\) is the final momentum, and \(\mathbf{p}_i\) is the initial momentum. If the mass (\textbf{m}) of the object remains constant—as it does in most practical scenarios—the equation simplifies to: \[ \mathbf{∆p} = m \cdot (\mathbf{v}_f - \mathbf{v}_i) \]
This calculation becomes much simpler when you realize that the momentum change is directly proportional to the change in velocity. So, if an object's velocity reverses direction, the change in momentum is essentially double the object's initial momentum in one direction.

Physics Problem Solving

Solving physics problems effectively requires not only an understanding of the concepts but also a systematic approach. For problems involving momentum, it's often helpful to break down the problem into these key steps:

Identify the Given Information

Start by listing out what's provided in the problem, like masses, velocities, and directions.

Determine What You're Solving For

Clarify your goal. Is it to find the final velocity, the change in momentum, or perhaps the force applied during a collision?

Apply the Concepts and Formulas

Use the relevant equations, such as the momentum formula and the change in momentum formula, to express what you're looking for in terms of the given variables.

Calculate

Execute the math involved; this could be arithmetic, algebra, or calculus, depending on the complexity of the problem.

Check Your Answers

Verify that your solution makes sense both mathematically and physically. Does it respect the principles like the conservation of momentum? Is the direction of the final momentum consistent with the described situation?

Applying this structured method enriches your understanding of the physics involved and enhances your problem-solving skills.