Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 37
To warm up for a match, a tennis player hits the \(57.0 \mathrm{~g}\) ball vertically with her racket. If the ball is stationary just before it is hit and goes \(5.50 \mathrm{~m}\) high, what impulse did she impart to it?
Short Answer
Expert verified
The impulse imparted to the ball is approximately 0.591 Ns.
Step by step solution
01
Identify the Known Values
We are given the mass of the tennis ball: \( m = 57.0 \text{ g} = 0.057 \text{ kg} \) (since we need to convert grams to kilograms for standard units), and the height the ball reaches: \( h = 5.50 \text{ m} \). The initial speed \( u = 0 \text{ m/s} \) because the ball is stationary before being hit.
02
Calculate the Final Velocity at Maximum Height
At the maximum height of \(5.50 \text{ m}\), the ball momentarily comes to rest, so its final velocity is \( v = 0 \text{ m/s} \). However, we need to find the initial velocity just after the hit using energy considerations. Use the equation for potential energy: \( mgh = \frac{1}{2}mv^2 \), where \( g = 9.81 \text{ m/s}^2 \).
03
Solve for Initial Velocity
Rearranging the potential energy equation: \[ v^2 = 2gh \] \[ v = \sqrt{2gh} \] Plugging in the known values, \[ v = \sqrt{2 \times 9.81 \times 5.50} \approx 10.37 \text{ m/s} \] gives the initial velocity of the ball after being hit.
04
Calculate the Impulse
Impulse is the change in momentum of the tennis ball and can be calculated using the formula: \[ \text{Impulse} = mv - mu \] Here, \( u = 0 \), so \[ \text{Impulse} = m \cdot v \] Substitute the known values: \[ \text{Impulse} = 0.057 \times 10.37 \approx 0.591 \, \text{Ns} \]
05
Conclude the Calculation
Therefore, the impulse imparted to the ball by the tennis racket is approximately \(0.591 \, \text{Ns}\).
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 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Physics Calculations
Physics calculations are the backbone of solving problems in physics, allowing us to quantify the behavior of matter and energy. In this specific exercise, we are dealing with a tennis ball and trying to determine the impulse exerted on it. To calculate such values effectively, it is important to identify known values such as initial speed, mass, and the height to which the ball is hit.
Here are steps commonly followed in physics calculations:
Here are steps commonly followed in physics calculations:
- Identifying Known Values: Gather all given information such as mass and initial velocity. For example, in this exercise, the tennis ball's mass is converted from grams to kilograms for standard unit calculations.
- Utilizing Formulas: Implement relevant formulas such as those from energy conservation and motion equations to derive unknown variables.
- Simplifying Equations: Break down equations into manageable steps to solve for the unknowns, like velocity or impulse in this scenario.
- Analyzing Results: Confirm the results make sense in the context of the problem.
Momentum and Impulse
Momentum is a fundamental concept in physics defined as the product of an object's mass and velocity. It signifies the quantity of motion an object possesses. Impulse, on the other hand, refers to the change in momentum resulting from a force applied over a period.
In the context of the exercise, the impulse given to the ball by the racket can be calculated using the change in momentum:
In the context of the exercise, the impulse given to the ball by the racket can be calculated using the change in momentum:
- The initial momentum: Since the ball is stationary initially, its initial momentum is zero.
- The change in momentum: Calculated when the racket hits the ball, giving it a velocity.
- The impulse formula: Impulse is the product of mass and velocity (since initial velocity is zero), calculated as \( \text{Impulse} = m \cdot v \).
Kinematic Equations
Kinematic equations are crucial for describing the motion of objects, especially those with constant acceleration. They relate velocity, acceleration, displacement, and time in uniformly accelerated motion, making them essential for this exercise.
In the exercise discussed, we use the concept of potential energy linked to the kinematic aspects of motion. Here鈥檚 how kinematic equations play a role:
In the exercise discussed, we use the concept of potential energy linked to the kinematic aspects of motion. Here鈥檚 how kinematic equations play a role:
- Maximum Height: At the peak of its trajectory, the ball momentarily stops, which means final velocity at that point is zero. This is derived from energy principles but is a kinematic consideration.
- Initial Velocity: We calculate the velocity just after the ball is hit using energy conservation. The equation \( v^2 = 2gh \), though primarily derived from conservation principles, ties into kinematics as it describes how the velocity changes with displacement.
Energy Conservation
The principle of energy conservation states that the total energy in an isolated system remains constant. It can be converted from one form to another but can't be created or destroyed.
In this exercise, energy conservation helps calculate the velocity of the ball after being struck. The tennis ball鈥檚 kinetic energy transforms into potential energy at its peak height:
In this exercise, energy conservation helps calculate the velocity of the ball after being struck. The tennis ball鈥檚 kinetic energy transforms into potential energy at its peak height:
- Kinetic Energy (KE): Initially, the ball possesses kinetic energy due to the motion imparted by the racket. KE is given by \( \frac{1}{2} mv^2 \).
- Potential Energy (PE): As the ball rises, kinetic energy is converted into potential energy at the peak height, calculated as \( mgh \).
- Energy Transformation: By equating kinetic and potential energy \( mgh = \frac{1}{2} mv^2 \), we can solve for the velocity needed to calculate impulse.
