Problem 44 - College Physics

Question

Two balls of equal mass \((0.50 \mathrm{~kg})\) approach the origin along the positive \(x\) - and \(y\) -axes at the same speed \((3.3 \mathrm{~m} / \mathrm{s})\). (a) What is the total momentum of the system? (b) Will the balls necessarily collide at the origin? What is the total momentum of the system after both balls have passed through the origin?

Step-by-Step Solution

Answer

(a) Total momentum is \( (1.65, 1.65) \) kg·m/s. (b) Yes, they will collide at the origin; total momentum after is \( (1.65, 1.65) \) kg·m/s.

Step 1: Calculate Momentum of Each Ball

Momentum is given by the formula \( p = mv \), where \( m \) is mass and \( v \) is velocity. Since both balls have a mass of 0.50 kg and are moving with a speed of 3.3 m/s, their momenta are \( p_x = 0.50 \times 3.3 = 1.65 \) kg·m/s for the ball on the x-axis, and \( p_y = 0.50 \times 3.3 = 1.65 \) kg·m/s for the ball on the y-axis.

Step 2: Determine Total Momentum of the System

Since momentum is a vector quantity, we need to consider both magnitude and direction. The ball moving along the x-axis has momentum \( (1.65, 0) \) kg·m/s and the ball moving along the y-axis has momentum \( (0, 1.65) \) kg·m/s. Therefore, the total momentum of the system is the vector sum \( (1.65, 0) + (0, 1.65) = (1.65, 1.65) \) kg·m/s.

Step 3: Examine Conditions for Collision at the Origin

For the balls to collide exactly at the origin, their paths need to intersect at the same time. Both balls travel the same distance to the origin at the same speed (3.3 m/s), so they will indeed collide at the origin if released simultaneously.

Step 4: Analyze Total Momentum After Passing Through the Origin

Momentum is conserved in the system. Once the balls pass through the origin, they will continue their motion along the x and y axes respectively. Therefore, the momentum after passing through the origin remains \( (1.65, 1.65) \) kg·m/s, the same as before they passed through the origin.

Key Concepts

Momentum

Momentum is a core concept in physics that helps us understand motion. At its simplest, momentum is the product of the mass and velocity of an object. Mathematically, it's expressed as \( p = mv \), where \( p \) stands for momentum, \( m \) is mass, and \( v \) is velocity.

• Mass: This refers to how much matter is in the object. In our exercise, both balls have a mass of 0.50 kg.
• Velocity: This is the speed of the object in a specific direction. Here, both balls travel at 3.3 m/s.
• Momentum Calculation: Using the formula, the momentum for each ball individually is 1.65 kg·m/s.

Momentum helps predict how an object will behave when it encounters other forces or objects. It's important for solving collision-related problems.

Vector Quantity

Momentum is not just a straightforward value; it's a vector quantity. This means that momentum has both magnitude and direction. Unlike scalar quantities (which have only size), vectors need directions too.

• X-axis Momentum: For the ball on the x-axis, its momentum is \( (1.65, 0) \) kg·m/s.
• Y-axis Momentum: For the ball moving along the y-axis, the momentum is \( (0, 1.65) \) kg·m/s.
• Total Momentum: You add these vectors using their components to find the total momentum. In this case, it's \( (1.65, 1.65) \) kg·m/s.

Understanding that momentum is a vector allows us to solve complex physics problems involving motion from multiple directions.

Collision

A collision occurs when two or more objects hit each other. In physics, analyzing collisions helps us understand how objects interact. Collisions can change the velocity, direction, and even shape of the involved objects.
In our exercise, both balls meet at the origin, and they travel at the same speed over equal distances. This synchrony leads to their collision.

• Equal Mass and Velocity: The fact that both balls have equal masses and velocities simplifies the calculation as they possess identical momentum values.
• Path Intersection: Since both balls reach the origin simultaneously, they will collide there.

Understanding collisions is crucial, as it involves momentum conservation to predict post-collision motion.

Physics Problem Solving

Physics problem-solving often involves breaking down a complex situation into simpler parts. The core strategy is to apply fundamental principles, like conservation of momentum, systematically.
For the problem at hand, we used several steps:

• Calculate Individual Momentums: By finding each ball's momentum separately, we use the basic formula for momentum.
• Analyze Total Momentum: By combining them into a vector sum, we get the system's total momentum before the collision.
• Check Collision Conditions: Ensuring the paths intersect at the right time calculated when they collide at the origin.
• Conservation Post-Collision: Recognizing that post-collision, the total momentum of the system remains unchanged at \( (1.65, 1.65) \) kg·m/s.

Physics is all about understanding the rules and applying them to predict how things behave. With practice, solving problems becomes intuitive.