Question

A cannonball of mass \(5.99 \mathrm{~kg}\) is shot from a cannon at an angle of \(50.21^{\circ}\) relative to the horizontal and with an initial speed of \(52.61 \mathrm{~m} / \mathrm{s}\). As the cannonball reaches the highest point of its trajectory, what is the gain in its potential energy relative to the point from which it was shot?

Short answer

Answer: The gain in potential energy relative to the point from which the cannonball was shot is approximately 4844.57 Joules.

Step-by-step solution

Identify the given information.

We are given: - Mass of the cannonball, which is \(m = 5.99\,\text{kg}\) - Launch angle, which is \(\theta = 50.21^\circ\) - Initial speed of the cannonball, which is \(v_0 = 52.61\,\text{m/s}\) - Gravitational acceleration, which is \(g = 9.81\,\text{m/s}^2\)

Calculate the vertical component of the initial velocity.

To find the maximum height, we first need the vertical component of the initial velocity, which can be found using the sine function and the initial speed: \(v_{0_y} = v_0 \sin \theta\) Plug in the given values: \(v_{0_y} = 52.61\,\text{m/s} \times \sin(50.21^\circ) \approx 40.27\,\text{m/s}\)

Calculate the time to reach maximum height.

The time it takes for the cannonball to reach its maximum height can be calculated by finding when its vertical velocity is 0. We can use the following formula: \(t = \frac{v_{0_y}}{g}\) Plug in the values we found: \(t = \frac{40.27\,\text{m/s}}{9.81\,\text{m/s}^2} \approx 4.10\,\text{s}\)

Calculate the maximum height.

To find the maximum height, we can use the vertical component of the initial velocity and the time to reach the maximum height: \(h = v_{0_y} t - \frac{1}{2}gt^2\) Plug in the values we found: \(h = 40.27\,\text{m/s} \times 4.10\,\text{s} - \frac{1}{2} \times 9.81\,\text{m/s}^2 \times (4.10\,\text{s})^2 \approx 82.11\,\text{m}\)

Calculate the gain in potential energy.

Finally, we can calculate the gain in potential energy at the highest point using the mass of the cannonball, the maximum height, and the gravitational acceleration: \(\Delta PE_p = mgh\) Plug in the given values and calculated height: \(\Delta PE_p = 5.99\,\text{kg} \times 9.81\,\text{m/s}^2 \times 82.11\,\text{m} \approx 4844.57\,\text{J}\) The gain in potential energy relative to the point from which the cannonball was shot is approximately 4844.57 Joules.

Key concepts

Understanding Kinematics

Kinematics is the branch of mechanics that focuses on the motion of objects without considering the forces that cause or result from the motion. When we talk about projectile motion, such as a cannonball being fired from a cannon, kinematics allows us to predict where and when the object will be on its path.

In our exercise, by knowing the initial speed and the angle of launch, we can separate the cannonball's initial velocity into horizontal and vertical components. It's crucial because these components are independent of each other and describe the motion along the x-axis (horizontal) and y-axis (vertical) separately. Using kinematic equations, we can find important information such as the time it takes to reach the maximum height (the top point on its trajectory), and from here we can determine other properties like the maximum height itself or the range.

By dissecting motion into these components, students can clearly see how the combination of horizontal motion (at a constant speed since there's no air resistance in ideal kinematic problems) and vertical motion (affected by gravity) shape the trajectory of a projectile. This approach simplifies complex motion into manageable calculations, making it easier to grasp for learners.

Potential Energy in Projectile Motion

Potential energy, particularly gravitational potential energy, refers to the energy that an object possesses due to its position in a gravitational field. Generally, the higher the object is above a reference point, the greater its potential energy.

Using the formula \(\Delta PE_p = mgh\), where 'm' stands for mass, 'g' for the acceleration due to gravity, and 'h' for height, we can calculate the gain in potential energy. In the context of our cannonball problem, as it ascends, it loses kinetic energy but gains gravitational potential energy, reaching a maximum when the cannonball is at its highest point.

The concept of energy conservation states that energy can neither be created nor destroyed, only converted from one form to another. So, for a projectile, the loss in kinetic energy as it rises is exactly equal to the gain in potential energy when frictional forces like air resistance are negligible. This correlation provides a critical link between the kinematic aspects of the motion and the energy perspective, presenting students with a more complete understanding of projectile motion dynamics.

Trajectory of a Projectile

The trajectory of a projectile is the path that it follows as it moves through space under the influence of gravity. This path is typically a parabola in shape because the horizontal component of motion is uniform, while the vertical component is subject to gravitational acceleration, pulling the object down.

In our exercise, the cannonball follows a curved path from the moment it leaves the cannon until it impacts the ground. At launch, the cannonball has a certain amount of kinetic energy that is gradually converted to potential energy as it rises, eventually stopping momentarily at its peak height where the kinetic energy is zero and potential energy is at its maximum. As it falls back down, the process reverses. The visual representation of this process helps students envision the combined effects of gravity and initial launch conditions.

Understanding the factors affecting a projectile's trajectory—such as launch angle, initial speed, and gravity—enables students to predict landing spots and optimize for distance or height. This fundamental concept in physics not only explains everyday phenomena, like a ball thrown in the air, but also has practical applications in various fields including sports, engineering, and even space exploration.