Chapter 7: Problem 5
A baseball is thrown from the roof of a 22.0-m-tall building with an initial velocity of magnitude 12.0 m/s and directed at an angle of 53.1\(^\circ\) above the horizontal. (a) What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. (b) What is the answer for part (a) if the initial velocity is at an angle of 53.1\(^\circ\) \(below\) the horizontal? (c) If the effects of air resistance are included, will part (a) or (b) give the higher speed?
Short Answer
Step by step solution
Understand the problem
Set up energy conservation equation
Calculate components of initial velocity
Apply energy conservation (part a)
Simplify and solve for final speed (part a)
Evaluate part b
Analyze air resistance impact (part c)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Projectile Motion
In this exercise, the baseball is thrown from a rooftop at a specified angle either above or below the horizontal. The initial velocity has to be broken down into two components:
- Horizontal component (\( v_{ix} \)): This doesn't change because gravity only affects the vertical component.
- Vertical component (\( v_{iy} \)): This is affected by gravity, causing the projectile to slow down as it rises and speed up as it falls towards the ground.
Kinetic Energy
In this problem, the initial kinetic energy comes from the baseball being thrown with a known initial velocity. As the ball moves through its trajectory, its speed changes, so its kinetic energy also changes. When it reaches its maximum height, the vertical component of the velocity becomes zero, reducing the kinetic energy. However, as it falls back down, the speed increases, and so does its kinetic energy.
One key point about kinetic energy and projectile motion is that while the speed (and thus kinetic energy) changes throughout the motion, the total mechanical energy of the system is conserved, assuming no air resistance. This means the energy shifts between kinetic and potential forms but remains constant overall.
Potential Energy
Initially, when the baseball is on the roof, it has a specific amount of potential energy owing to its height above the ground level. As it is thrown into the air, its potential energy increases when ascending until it reaches its peak, and then decreases as it descends. At the highest point, potential energy is at a maximum, while kinetic energy is minimized. Conversely, just before the projectile hits the ground, its potential energy is minimized, and kinetic energy is maximized.
The conservation of mechanical energy principle helps us understand that even as potential and kinetic energies trade places throughout the motion, their sum remains constant, assuming no external forces (like air resistance) act on the projectile. This is crucial in predicting the motion and behavior of the projectile accurately.