/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 A cannonball of mass \(5.99 \mat... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 38

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

Expert verified
Answer: To calculate the gain in potential energy, follow these steps: 1. Calculate the vertical component of the initial velocity using the formula: \(v_{0y} = v_0 \times \sin(\theta)\). 2. Convert the angle from degrees to radians using the formula: \(\theta(rad) = \frac{\theta(deg) \times \pi}{180}\). 3. Calculate the time taken to reach the highest point using the formula: \(t = \frac{v_{0y}}{g}\). 4. Calculate the maximum height above the ground using the formula: \(h = v_{0y}t - \frac{1}{2}gt^2\). 5. Calculate the potential energy gained using the formula: \(\Delta PE = mgh\). By following these steps, you will find the gain in potential energy at the highest point of the cannonball's trajectory.

Step by step solution

01

Calculate the vertical component of the initial velocity

Using the angle of projection (50.21掳) and the initial speed (52.61 m/s), we can determine the initial vertical velocity component. We use the trigonometric sine function: $$ v_{0y} = v_0 \times \sin(\theta) $$ where \(v_0\) is the initial speed (52.61 m/s), \(\theta\) is the angle (50.21掳), and \(v_{0y}\) is the vertical component of the velocity.
02

Convert the angle from degrees to radians

To use the sine function, we need to convert the angle from degrees to radians: $$ \theta(rad) = \frac{\theta(deg) \times \pi}{180} $$
03

Calculate the time taken to reach the highest point

The time taken to reach the highest point can be calculated using the kinematic formula: $$ v_y = v_{0y} - gt $$ At the highest point, the vertical velocity (\(v_y\)) will be 0. So solving for \(t\), where \(g\) is the gravitational acceleration (9.81 m/s虏): $$ t = \frac{v_{0y}}{g} $$
04

Calculate the maximum height above the ground

Use the time calculated above to find the maximum height reached by the cannonball: $$ h = v_{0y}t - \frac{1}{2}gt^2 $$
05

Calculate the potential energy gained

The gain in potential energy is found by multiplying the mass, gravitational acceleration, and the maximum height: $$ \Delta PE = mgh $$ Where \(m\) is the mass of the cannonball (5.99 kg), \(g\) is gravitational acceleration (9.81 m/s虏), and \(h\) is the maximum height.

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