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Chapter 5: Problem 62

An outfielder throws a \(0.150-\mathrm{kg}\) baseball at a speed of \(40.0 \mathrm{~m} / \mathrm{s}\) and an initial angle of \(30.0^{\circ}\). What is the kinetic energy of the ball at the highest point of its motion?

Short Answer

Expert verified
The kinetic energy of the baseball at the highest point of its motion is \(90.0 \mathrm{J}\).

Step by step solution

01

Find the horizontal component of initial velocity

The horizontal component of velocity \(v_x\) can be found using the formula \(v_x = v \cdot \cos(\theta)\), where \(v=40.0 \mathrm{~m/s}\) is the initial speed and \(\theta=30.0^{\circ}\) is the angle. First convert the angle from degrees to radians, because the cosine function in calculators uses radians, not degrees. Then substitute the known values to get \(v_x = 40.0 \mathrm{m/s} \cdot \cos(30.0 \cdot \frac{\pi}{180})\).
02

Calculate the horizontal component of initial velocity

After doing the calculation from the previous step, it turns out that the horizontal component of initial velocity \(v_x\) is equal to \(34.64 \mathrm{m/s}\).
03

Find the kinetic energy at the highest point

At the highest point, the kinetic energy can be found using the formula \(KE = \frac{1}{2} m v_x^{2}\), where \(m = 0.150 \mathrm{kg}\) is the mass of the ball. Substitute the known values to get \(KE = \frac{1}{2} \cdot 0.150 \mathrm{kg} \cdot (34.64 \mathrm{m/s})^{2}\).
04

Calculate the kinetic energy at the highest point

After doing the calculation from the previous step, it turns out that the kinetic energy of the baseball at the highest point of its motion is about \(90.0 \mathrm{J}\).

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Key Concepts

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

Projectile Motion
Projectile motion occurs when an object is thrown into the air and moves along a curved path under the influence of gravity. In our problem, the baseball thrown by the outfielder follows this path. This type of motion is divided into two components: horizontal and vertical. These movements are independent of each other but happen simultaneously.
  • Horizontal motion: Constant velocity, as there is no acceleration (ignoring air resistance).
  • Vertical motion: Affected by gravity, causing the object to accelerate downwards.
Understanding these components helps in analyzing the trajectory of the projectile and answering questions like the maximum height, range, and time of flight.
At the highest point, the vertical component of the velocity becomes zero, but the horizontal component remains unchanged. That's why we use it to calculate the kinetic energy there.
Physics Problem Solving
When tackling physics problems like this, it's crucial to have a systematic approach. Start by identifying what is being asked and what information you already have. In this case, we're looking at the kinetic energy of the baseball at its highest point. We break the problem down into clear, manageable steps.

Analyzing Given Data:

  • The initial velocity is given, making it possible to find its components.
  • The mass of the baseball is known, which is essential for calculating kinetic energy.

Proceed to find the horizontal component of the velocity using trigonometric functions. This leads to calculating the kinetic energy using the standard formula for kinetic energy \( KE = \frac{1}{2} mv^2 \). A step-by-step breakdown ensures accurate computations and clarity in solving similar problems.
Horizontal and Vertical Components of Velocity
Understanding the horizontal and vertical components of velocity is vital in solving projectile motion problems. The initial velocity of the projectile can be divided into these two components using trigonometric functions such as sine and cosine.

Calculating Components:

  • Horizontal Component: Calculated using \( v_x = v \cdot \cos(\theta) \). This remains constant throughout the projectile's flight (ignoring air resistance).
  • Vertical Component: Found using \( v_y = v \cdot \sin(\theta) \). Influenced by gravity, it determines the maximum height and the time the projectile stays in the air.

For our baseball problem, we only needed the horizontal component to find the kinetic energy at the highest point. Recognizing when and how to focus on each component simplifies the problem-solving process, allowing you to target and calculate specific elements of motion accurately.

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Most popular questions from this chapter

GP A \(60.0-\mathrm{kg}\) athlete leaps straight up into the air from a trampoline with an initial speed of \(9.0 \mathrm{~m} / \mathrm{s}\). The goal of this problem is to find the maximum height she attains and her speed at half maximum height. (a) What are the interacting objects and how do they interact? (b) Select the height at which the athlete's speed is \(9.0 \mathrm{~m} / \mathrm{s}\) as \(y=0\). What is her kinetic energy at this point? What is the gravitational potential energy associated with the athlete? (c) What is her kinetic energy at maximum height? What is the gravitational potential energy associated with the athlete? (d) Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer. (c) Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.

An older-model car accelerates from 0 to speed \(v\) in 10 s. A newer, more powerful sports car of the same mass accelerates from 0 to \(2 v\) in the same time period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars.

A toy gun uses a spring to project a \(5.3-g\) soft rubber sphere horizontally. The spring constant is \(8.0 \mathrm{~N} / \mathrm{m}\), the barrel of the gun is \(15 \mathrm{~cm}\) long, and a constant frictional force of \(0.032 \mathrm{~N}\) exists between barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed \(5.0 \mathrm{~cm}\) for this launch?

A 650-kg elevator starts from rest and moves upward for \(3.00 \mathrm{~s}\) with constant acceleration until it reaches its cruising speed, \(1.75 \mathrm{~m} / \mathrm{s}\). (a) What is the average power of the elevator motor during this period? (b) How does this amount of power compare with its power during an upward trip with constant speed?

BIO (a) A 75-kg man steps out a window and falls (from rest) \(1.0 \mathrm{~m}\) to a sidewalk. What is his speed just before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for his fall is an approximately \(0.50-\mathrm{cm}\) give in the pads of his feet. Calculate the average force exerted on him by the ground during this \(0.50 \mathrm{~cm}\) of travel. This average force is sufficient to cause damage to cartilage in the joints or to break bones.
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