Problem 19 - College Physics

Question

When tossed upward and hit horizontally by a batter, a \(0.20-\mathrm{kg}\) softball receives an impulse of \(3.0 \mathrm{~N} \cdot \mathrm{s}\). With what horizontal speed does the ball move away from the bat?

Step-by-Step Solution

Answer

The ball moves away with a horizontal speed of 15.0 m/s.

Step 1: Understanding Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can be calculated using the formula: \[ \text{Impulse} = \Delta p = m \times \Delta v \]where \(m\) is the mass of the object and \(\Delta v\) is the change in velocity.

Step 2: Setting Up the Equation

Given that the impulse is \(3.0\, \mathrm{N} \cdot \mathrm{s}\) and the mass of the softball is \(0.20\, \mathrm{kg}\), we can set up the equation for impulse as follows:\[ 3.0 = 0.20 \times \Delta v \] Here, \(\Delta v\) is the change in horizontal velocity of the ball due to the impulse.

Step 3: Solving for the Change in Velocity

To find the change in velocity \(\Delta v\), we need to isolate it in the equation:\[ \Delta v = \frac{3.0}{0.20} \]Perform the division to solve for \(\Delta v\).

Step 4: Calculating the Horizontal Speed

Calculate the value:\[ \Delta v = \frac{3.0}{0.20} = 15.0 \]Therefore, the change in the horizontal speed of the ball is \(15.0\, \mathrm{m/s}\).

Key Concepts

Impulse and Momentum

Impulse and momentum are tightly linked concepts in physics, particularly when analyzing collisions and interactions between objects. When a force acts on an object over a period of time, it results in an impulse. Impulse is the product of the force and the time duration over which it acts. An important characteristic of impulse is its effect on an object's momentum. Momentum, which can be thought of as the "quantity of motion," is defined as the product of an object's mass and its velocity. In equation form, impulse is the change in momentum: \[ \text{Impulse} = \Delta p = m \times \Delta v \]where \(m\) represents the mass and \(\Delta v\) is the change in velocity.
Impulse is a vector quantity, meaning it has both direction and magnitude. This feature becomes particularly relevant in problems involving changes in direction, such as a softball being hit by a bat. Whether or not the original velocity is significant, impulse can indicate how fast and in which direction the object will move post-collision.

Physics Calculations

Physics calculations often rely on understanding and manipulating mathematical equations to uncover values of interest. In the context of impulse, the equation \( \Delta p = m \times \Delta v \) can help us calculate the resultant velocity of an object after an impulse has been applied.
To approach such problems, start by identifying the known values:

• Mass of the object (\(m\))
• Impulse applied (\(\text{Impulse}\))

Use these values to solve for the unknown, often the change in velocity \((\Delta v)\). Rearrange the formula to isolate \(\Delta v\):

• \( \Delta v = \frac{\text{Impulse}}{m} \)

By dividing the given impulse by the mass of the object, the resultant change in velocity can be found. This step requires precise arithmetic to ensure accurate results, considering units and decimal placement for consistency in measurements. In our example, this helps us find out how fast the softball moves after being hit.

Kinematics

Kinematics is a branch of physics that describes the motion of objects without necessarily considering the causes of motion such as forces. However, it is essential to understand to effectively solve momentum and impulse problems as part of the bigger physics framework.
In the example of a softball being hit, kinematics will focus on how the velocity changes due to the impulse imparted by the bat. The main parameter here is the horizontal speed of the ball post-collision, determined through a series of calculations stemming from the impulse-mass relationship.
Key aspects of kinematic calculations include:

• Interpreting the relationship between velocity changes and external forces
• Understanding vectors as speeds might involve both magnitude and direction
• Using derived values like \(\Delta v\) to find subsequent positions, speeds, etc.

This focus on movement dynamics underpins more complex physics scenarios, where initial conditions like throwing angles or bat impact points might complicate the analysis, but can still be approached sequentially through kinematic principles.