91Ó°ÊÓ

Two balls of equal mass approach the coordinate origin, one moving downward along the \(y\) -axis at \(2.00 \mathrm{~m} / \mathrm{s}\) and the other moving to the right along the \(-x\) -axis at \(3.00 \mathrm{~m} / \mathrm{s}\). After they collide, one ball moves out to the right along the \(+x\) -axis at \(1.20 \mathrm{~m} / \mathrm{s}\). Find the scalar \(x\) and \(y\) velocity components of the other ball. This is a two-dimensional collision and momentum must be conserved independently in each perpendicular direction, \(x\) and \(y\). Take \(u p\) and to the right as positive. Accordingly, keeping in mind that before impact only one ball had an \(x\) -component of velocity, Or $$ \begin{aligned} \text { (Momentum before) }_{x} &=\text { (Momentum after) }_{x} \\ m(3.00 \mathrm{~m} / \mathrm{s})+0 &=m(1.20 \mathrm{~m} / \mathrm{s})+m v_{x} \end{aligned} $$ Here \(v_{\mathrm{x}}\) is the unknown \(x\) -component of velocity of the second ball acquired on impact. Since we know that the first ball lost some of its \(x\) -momentum, the second ball must have gained it. Moreover, (Momentum before) \(_{y}=(\text { Momentum after })_{y}\) or $$ 0+m(-2.00 \mathrm{~m} / \mathrm{s})=0+m v_{y} $$ Here \(v_{\mathrm{y}}\) is the \(y\) -component of velocity of the second ball. (Why the minus sign?) Solving each equation, after cancelling the mass we find that \(v_{x}=1.80 \mathrm{~m} / \mathrm{s}\) and \(v_{y}=-2.00 \mathrm{~m} / \mathrm{s}\)

Step by step solution

01

Understand the Problem

We have two balls of equal mass colliding. One ball is moving along the -x-axis and the other along the y-axis. We need to apply the conservation of momentum in the x and y directions to find the velocity components of the second ball after the collision.

02

Set Up the X-direction Conservation of Momentum Equation

Before the collision, the momentum in the x-direction is due only to the first ball: \[ m(3.00 \text{ m/s}) + 0 = \text{Momentum before}_{x} \]After the collision, the momentum in the x-direction is shared by both balls: \[ m(1.20 \text{ m/s}) + m v_{x} = \text{Momentum after}_{x} \]Using these, the equation becomes:\[ m(3.00 \text{ m/s}) = m(1.20 \text{ m/s}) + m v_{x} \]

03

Solve for the X-component of Velocity

By cancelling the mass \(m\) from the equation, we solve for \(v_{x}\):\[ 3.00 \text{ m/s} = 1.20 \text{ m/s} + v_{x} \]\[ v_{x} = 3.00 \text{ m/s} - 1.20 \text{ m/s} = 1.80 \text{ m/s} \]

04

Set Up the Y-direction Conservation of Momentum Equation

Before the collision, only the second ball has momentum in the y direction:\[ 0 + m(-2.00 \text{ m/s}) = \text{Momentum before}_{y} \]After the collision, the momentum in the y direction for the first ball is zero, so:\[ 0 = m (-2.00 \text{ m/s}) + m v_{y} \]

05

Solve for the Y-component of Velocity

By cancelling the mass \(m\) from the equation, we solve for \(v_{y}\):\[ -2.00 \text{ m/s} = v_{y} \]

06

Conclusion: Combine the Results

The second ball's velocity components after the collision are \(v_{x} = 1.80 \text{ m/s}\) to the right and \(v_{y} = -2.00 \text{ m/s}\) downward.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Two-dimensional Collision

In two-dimensional collisions, objects move and interact in a plane, involving components of motion in both the x and y directions. This kind of collision occurs when objects not only move forward but also side-to-side, resulting in a need to analyze two separate dimensions. Understanding two-dimensional collisions requires looking at momentum separately along these axes.

*Remember:* The principle of momentum conservation tells us that the total momentum in a system remains the same before and after a collision. Therefore, for a comprehensive analysis, it's crucial to apply this principle independently to each direction:

• **X-axis collision:** Accounts for horizontal motion. Object paths that are horizontally aligned mainly affect this axis.
• **Y-axis collision:** Handles vertical movement. Any vertical shift, including drops or lifts, plays here.

Recognizing the separate impacts in each direction helps predict post-collision velocities accurately.

Velocity Components

Velocity components break down an object's movement into vertical and horizontal parts, simplifying vector analysis of motion. In physics problems like two-dimensional collisions, focusing on velocity components allows us to apply momentum conservation principles more effectively. We can clearly see how each part of the velocity contributes to the overall motion.

When resolving velocity into components, use the following steps:

• **Identify directions:** Determine which axis represents what direction. Usually, the x-axis is horizontal, and the y-axis is vertical.
• **Analyze individual components:** Each axis isolates part of the velocity, helping separate calculations for horizontal and vertical momentum.

This approach simplifies complex problems by allowing separate calculations for each directional effect, ensuring a holistic view of motion dynamics.

Physics Problem Solving

Physics problem-solving, particularly involving momentum and collisions, demands logical structuring and careful analysis. The key is to break down questions into smaller, manageable parts. By doing so, complex entity interactions become more understandable.

Here’s a straightforward approach to tackle physics problems:

• Understand the Problem: Begin by identifying the given information and what the problem asks for. This is critical in constructing a strategy, as was done using momentum conservation laws.
• Formulate Equations: Create mathematical representations of conditions and conservation principles relevant to the scenario. Equations make abstract ideas more concrete and manageable.
• Apply Logical Steps: Solve systematically, addressing one component at a time, be it x or y direction. Each step resolves a piece of the overall problem, building towards a comprehensive solution.

By approaching systematically, even the most challenging situations can be dissected efficiently, leading to accurate results.

Kinematics

Kinematics studies the motion of objects without considering the causes of motion. In context with momentum conservation, it focuses on analyzing speed and direction changes during collisions. It's the foundational science guiding many physics applications.

To understand kinematics, we look at factors such as:

• **Displacement:** Change in object position over time, critical in evaluating pre- and post-collision positions.
• **Velocity:** Speed with a directional component, affected during collisions but quantifiable using kinematic principles.
• **Acceleration:** Changes in velocity often occur, though not always relevant in instantaneous collision analysis.

Focusing on these elements helps decouple complex motion into understandable pieces, allowing for more precise conclusions about how objects interact within a system.