Problem 40 - College Physics

Question

\(\bullet\) To warm up for a match, a tennis player hits the 57.0 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?

Step-by-Step Solution

Answer

The impulse imparted to the tennis ball is approximately 0.592 kg m/s.

Step 1: Analyze the Problem

We are given a tennis ball with a mass of 57.0 g (which we will convert to 0.057 kg), which is hit vertically and reaches a height of 5.50 m. We need to calculate the impulse imparted to the ball. Impulse is the change in momentum of the object.

Step 2: Calculate the Final Velocity

First, we determine the speed of the tennis ball at its maximum height using the kinematic equation for vertical motion. Since the initial velocity at the top is zero, we use \[ v^2 = u^2 + 2as \]where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration due to gravity \(-9.81 \text{ m/s}^2\), and \( s \) is the height (5.50 m):\[ 0 = u^2 - 2 \times 9.81 \times 5.50 \]Solve for \( u \) to get \[ u = \sqrt{2 \times 9.81 \times 5.50} \approx 10.38 \text{ m/s} \].

Step 3: Determine the Change in Momentum

The change in momentum, which equals the impulse, is given by the formula \( \Delta p = m \Delta v \), where \( m \) is the mass of the tennis ball, and \( \Delta v \) is the change in velocity. Initially, the ball was at rest, so \( \Delta v = 10.38 \text{ m/s} \).\[ \Delta p = 0.057 \times 10.38 \approx 0.592 \text{ kg} \, \text{m/s} \].

Key Concepts

Understanding Impulse in Physics Problems

In physics, impulse is a significant concept that explains how the motion of an object changes due to a force applied over a specific time frame. It is directly related to changes in an object’s momentum. Mathematically, impulse is expressed as the product of the force applied and the time duration for which it acts on an object: \[ J = F \times \Delta t \].
Impulse also equals the change in momentum of the object. This can be formulated as \[ J = \Delta p = m \times \Delta v \], where \( m \) is mass and \( \Delta v \) is the change in velocity.

• When a tennis player hits a ball, she applies force over a short duration, imparting impulse.
• The ball's velocity changes from resting to launching speed, indicating impulse.

Understanding impulse helps in analyzing how forces affect objects in motion over time, crucial for solving various physics problems.

Kinematic Equations for Solving Motion Problems

Kinematic equations are fundamental tools in physics, particularly for solving problems involving motion. These equations relate the five key motion variables: displacement, initial velocity, final velocity, acceleration, and time. The typical kinematic equation is: \[ v^2 = u^2 + 2as \].
This equation is especially helpful when analyzing vertical motion, such as in the case of our tennis ball. Here, the initial velocity \( u \) was unknown, but the height \( s \) and gravitational acceleration \( a = -9.81 \text{ m/s}^2\) allowed us to find it.

• By using kinematic equations, we can determine the maximum speed of the ball using initial or final velocity and position data.
• This is a crucial step for understanding how forces influence the movement over time.

These equations provide a structured way to compute essential variables, answering many motion-related questions efficiently.

Momentum and Its Role in Physics

Momentum in physics signifies the "inertia in motion" of an object. It’s the product of an object's mass and its velocity, conveyed by the expression \[ p = m \times v \].
Momentum is a vector quantity, having both magnitude and direction. Changes in momentum are crucial when objects interact due to forces, as seen in our tennis ball example.

• In this scenario, the ball initially had zero momentum when stationary.
• The change in velocity when hit creates a new momentum, indicating motion commencement.

The law of conservation of momentum implies that in an isolated system, the total momentum before and after a force acts remains constant unless external forces are involved. This principle is vital for solving collision and interaction problems.

Vertical Motion and How It's Analyzed

Vertical motion refers to the movement of objects under the influence of gravity alone, often ignoring air resistance for simplicity. Falling freely or being projected upwards are examples of such motion patterns.
The tennis ball, when hit upwards, provides a classic illustration of vertical motion. Here’s how vertical motion can be assessed:

• Use gravitational acceleration \(-9.81 \text{ m/s}^2\) for downward motion.
• Calculate the maximum height using the kinematic equations.

By analyzing motion in the vertical plane, students will understand how objects behave under gravitational influence. This deepens comprehension of concepts like energy transformation and maximum height attained, all crucial for tackling real-world scenarios.