Midterm ExamScore: 0/100 Answered: 0/18---Question 1A Cannonball Is Launched Into The Air With An Upward Velocity Of 160 Feet Per Second From A Cannon On A High Mountain 896 Feet From The Ground. The Height \[$ H \$\] Of The Cannonball After

by ADMIN 243 views

Introduction

Projectile motion is a fundamental concept in physics that describes the motion of an object under the influence of gravity. In this problem, we will explore the trajectory of a cannonball launched into the air from a high mountain. The height of the cannonball after a certain time can be calculated using the equations of motion. In this article, we will derive the equation for the height of the cannonball and use it to calculate the height at a given time.

Equations of Motion

The equations of motion for an object under the influence of gravity are given by:

  • Horizontal motion: x(t) = x0 + v0xt
  • Vertical motion: y(t) = y0 + v0yt - (1/2)gt^2

where:

  • x(t) and y(t) are the positions of the object at time t
  • x0 and y0 are the initial positions of the object
  • v0x and v0y are the initial velocities of the object
  • g is the acceleration due to gravity (approximately 32 ft/s^2)

Cannonball Trajectory

In this problem, the cannonball is launched from a high mountain with an upward velocity of 160 feet per second. The initial position of the cannonball is 896 feet from the ground. We can use the equations of motion to calculate the height of the cannonball at a given time.

Derivation of the Equation for Height

Let's assume that the cannonball is launched at time t = 0. The initial position of the cannonball is y0 = 896 feet, and the initial velocity is v0y = 160 ft/s. We can use the equation for vertical motion to calculate the height of the cannonball at time t:

y(t) = y0 + v0yt - (1/2)gt^2

Substituting the values, we get:

y(t) = 896 + 160t - (1/2)(32)t^2

Simplifying the equation, we get:

y(t) = 896 + 160t - 16t^2

This is the equation for the height of the cannonball at time t.

Calculating the Height at a Given Time

Now that we have the equation for the height of the cannonball, we can use it to calculate the height at a given time. Let's say we want to calculate the height at time t = 2 seconds.

y(2) = 896 + 160(2) - 16(2)^2 y(2) = 896 + 320 - 64 y(2) = 1152

Therefore, the height of the cannonball at time t = 2 seconds is 1152 feet.

Graphing the Trajectory

The trajectory of the cannonball can be graphed using the equation for height. We can use a graphing calculator or a computer program to plot the graph.

Conclusion

In this article, we derived the equation for the height of a cannonball launched into the air from a high mountain. We used the equations of motion to calculate the height at a given time and graphed the trajectory of the cannonball. This problem demonstrates the application of projectile motion and the use of equations of motion to solve real-world problems.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics (9th ed.). John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers (10th ed.). Cengage Learning.

Additional Problems

  1. A particle is moving in a straight line with an initial velocity of 20 m/s. If the acceleration is 2 m/s^2, find the velocity and position of the particle after 5 seconds.
  2. A ball is thrown upward from the ground with an initial velocity of 25 m/s. If the acceleration due to gravity is 9.8 m/s^2, find the maximum height reached by the ball and the time it takes to reach the maximum height.
  3. A car is traveling at a speed of 60 km/h. If the driver applies the brakes and the car decelerates at a rate of 2 m/s^2, find the time it takes for the car to come to a stop.

Answer Key

  1. Velocity: 40 m/s, Position: 100 m
  2. Maximum height: 25 m, Time: 2.55 s
  3. Time: 30 s
    Midterm Exam: Projectile Motion and Cannonball Trajectory - Q&A ================================================================

Introduction

In our previous article, we explored the trajectory of a cannonball launched into the air from a high mountain. We derived the equation for the height of the cannonball and used it to calculate the height at a given time. In this article, we will answer some frequently asked questions related to projectile motion and the cannonball trajectory.

Q&A

Q1: What is the difference between horizontal and vertical motion?

A1: Horizontal motion refers to the motion of an object in a straight line, while vertical motion refers to the motion of an object under the influence of gravity. In the case of the cannonball, the horizontal motion is constant, while the vertical motion is affected by gravity.

Q2: How do you calculate the height of the cannonball at a given time?

A2: To calculate the height of the cannonball at a given time, you can use the equation for vertical motion:

y(t) = y0 + v0yt - (1/2)gt^2

where:

  • y(t) is the height of the cannonball at time t
  • y0 is the initial height of the cannonball
  • v0y is the initial velocity of the cannonball
  • g is the acceleration due to gravity
  • t is the time

Q3: What is the maximum height reached by the cannonball?

A3: The maximum height reached by the cannonball can be calculated using the equation for vertical motion. Since the cannonball is launched upward, the maximum height will occur when the vertical velocity is zero. We can use the equation:

v(t) = v0y - gt

to find the time at which the vertical velocity is zero. Then, we can use the equation for height to find the maximum height.

Q4: How do you graph the trajectory of the cannonball?

A4: To graph the trajectory of the cannonball, you can use a graphing calculator or a computer program to plot the equation for height:

y(t) = y0 + v0yt - (1/2)gt^2

You can also use a parametric plot to graph the trajectory of the cannonball.

Q5: What are some real-world applications of projectile motion?

A5: Projectile motion has many real-world applications, including:

  • Ballistics: Projectile motion is used to calculate the trajectory of projectiles, such as bullets and artillery shells.
  • Rocket science: Projectile motion is used to calculate the trajectory of rockets and spacecraft.
  • Sports: Projectile motion is used to calculate the trajectory of balls in sports, such as baseball and golf.
  • Engineering: Projectile motion is used to design and optimize the trajectory of projectiles, such as missiles and satellites.

Conclusion

In this article, we answered some frequently asked questions related to projectile motion and the cannonball trajectory. We hope that this article has provided you with a better understanding of the concepts and equations involved in projectile motion.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics (9th ed.). John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers (10th ed.). Cengage Learning.

Additional Resources

  • Online tutorials: There are many online tutorials and resources available that can help you learn about projectile motion and the cannonball trajectory.
  • Textbooks: There are many textbooks available that cover the topic of projectile motion and the cannonball trajectory.
  • Software: There are many software programs available that can help you simulate and visualize the trajectory of the cannonball.

Answer Key

  1. Horizontal motion refers to the motion of an object in a straight line, while vertical motion refers to the motion of an object under the influence of gravity.
  2. To calculate the height of the cannonball at a given time, you can use the equation for vertical motion: y(t) = y0 + v0yt - (1/2)gt^2
  3. The maximum height reached by the cannonball can be calculated using the equation for vertical motion.
  4. To graph the trajectory of the cannonball, you can use a graphing calculator or a computer program to plot the equation for height.
  5. Some real-world applications of projectile motion include ballistics, rocket science, sports, and engineering.