Problem 56 - University Physics with Modern Physics

Question

A ball is thrown upward with an initial velocity of 15 m/s at an angle of 60.0\(^\circ\) above the horizontal. Use energy conservation to find the ball's greatest height above the ground.

Step-by-Step Solution

Answer

The ball reaches a height of approximately 8.60 meters.

Step 1: Understand the Problem

We need to find the greatest height reached by the ball when thrown upwards at an angle of 60 degrees with an initial speed of 15 m/s. We will use the principle of conservation of energy to find this height.

Step 2: Define the Energy Conservation Equation

According to the conservation of mechanical energy, the initial kinetic energy (KE) of the ball will be converted into gravitational potential energy (PE) at its highest point. We have:\[\text{Initial } KE = \frac{1}{2} m v_i^2 \quad \text{and} \quad \text{Final } PE = mgh\]Where \(v_i\) is the initial vertical component of velocity, \(m\) is the mass of the ball, \(g\) is the acceleration due to gravity, and \(h\) is the unknown height.

Step 3: Calculate the Initial Vertical Velocity Component

The vertical component of the initial velocity \(v_{i, y}\) can be calculated as:\[v_{i, y} = v_i \cdot \sin(\theta)\]Where \(v_i = 15\, \text{m/s}\) and \(\theta = 60.0^{\circ}\). Calculating gives:\[v_{i, y} = 15 \cdot \sin(60^{\circ}) = 15 \cdot \frac{\sqrt{3}}{2} \approx 12.99\, \text{m/s}\]

Step 4: Relate Initial Kinetic and Final Potential Energy

Set the initial kinetic energy equal to the final potential energy:\[\frac{1}{2} m (v_{i, y})^2 = mgh\]Because the mass \(m\) is on both sides of the equation, it cancels out. We then have:\[\frac{1}{2} (12.99)^2 = gh\]

Step 5: Solve for the Height

We can now solve for \(h\) using the equation:\[h = \frac{(12.99)^2}{2 \times 9.81}\]Calculating this gives:\[h = \frac{168.7401}{19.62} \approx 8.60 \text{ m}\]

Step 6: Conclusion

The greatest height the ball reaches above the ground is approximately 8.60 meters. The calculations are based on converting all initial kinetic energy into potential energy at the highest point.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When the ball is thrown upwards, it starts with a certain amount of kinetic energy, which is determined by its initial velocity and mass. The formula for kinetic energy is:\[ KE = \frac{1}{2}mv^2 \]At the beginning, the ball has maximal kinetic energy because it is moving quickly. As an object moves, especially when it is thrown upwards, its kinetic energy decreases because its speed reduces until it brings the object to a temporary stop at its peak height.- **Initial Kinetic Energy**: This is the energy when the ball just left your hand. This is when the ball is set in motion with the initial velocity split into vertical and horizontal components.- **Conversion to Potential Energy**: Important to note is that as the ball rises, its kinetic energy converts into potential energy due to the height it gains.Understanding kinetic energy helps us track how energy is transferred and transformed during the ball's motion.

Potential Energy

Potential energy is energy stored due to an object's position or height. In our exercise, when the ball reaches its maximum height, all of its initial kinetic energy has been converted into potential energy. The formula for gravitational potential energy is:\[ PE = mgh \]- **Gravitational Potential Energy**: This type of potential energy increases as the ball rises higher in the air. Here, 'h' is the height above the ground, 'm' is the mass of the ball, and 'g' is the gravitational pull of the earth, which is approximately 9.81 m/s². - **Energy Conservation**: As energy is conserved, potential energy at the peak will equal the initial kinetic energy (when air resistance is neglected).- **Maximum Height**: At the maximum height, kinetic energy is zero, and potential energy is at its peak. Knowing potential energy helps us calculate how high the ball goes by determining how much energy is converted from kinetic to potential energy.

Vertical Velocity

Vertical velocity is the component of the initial velocity that influences the ball’s height. It is affected by the angle at which an object is thrown. For the ball thrown at an angle, the initial velocity has two components—horizontal and vertical.- **Vertical Component Calculation**: The formula to find the vertical component of the initial velocity is: \[ v_{i,y} = v_i \cdot \sin(\theta) \] Here, \(v_i\) is the initial velocity of 15 m/s and \(\theta\) is the angle of 60 degrees.- **Effect on Motion**: The vertical component influences how high the object will go. The larger this component, the higher the ball will reach.- **Stopping Point**: The vertical velocity decreases as the ball ascends, eventually reaching zero at the ball's peak height.Understanding vertical velocity is crucial because it allows us to calculate the maximum height using energy conservation. It directly affects how kinetic energy is distributed as the ball ascends.