Problem 100 - College Physics

Question

\(\bullet\) \(\bullet\) \(\bullet\) A ball is thrown upward with an initial velocity of 15 \(\mathrm{m} / \mathrm{s}\) 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's greatest height is approximately 8.60 meters.

Step 1: Identify Known Values

First, identify the known values from the problem. The initial velocity of the ball is \( v_i = 15 \, \text{m/s} \), and the angle of projection is \( \theta = 60.0^\circ \). We are looking for the maximum height, which we'll denote as \( h_{max} \).

Step 2: Resolve Initial Velocity into Components

Splitting the initial velocity into horizontal and vertical components using trigonometry. The vertical component is \( v_{iy} = v_i \sin \theta = 15 \sin 60.0^\circ = 15 \times \frac{\sqrt{3}}{2} = 12.99 \, \text{m/s} \).

Step 3: Apply Energy Conservation Principle

The principle of energy conservation states that the initial kinetic energy is equal to the potential energy at the highest point. The kinetic energy initially is \( \frac{1}{2} m v_{iy}^2 \), and the potential energy at the maximum height \( h \) is \( mgh \). Set these equal: \( \frac{1}{2} m v_{iy}^2 = mgh_{max} \).

Step 4: Simplify the Energy Equation

Cancel mass \( m \) from both sides of the equation to get \( \frac{1}{2} v_{iy}^2 = gh_{max} \).

Step 5: Solve for Maximum Height

Rearrange the equation \( \frac{1}{2} v_{iy}^2 = gh_{max} \) to find \( h_{max} = \frac{v_{iy}^2}{2g} \). Substitute \( g = 9.81 \, \text{m/s}^2 \) and \( v_{iy} = 12.99 \, \text{m/s} \) to get \( h_{max} = \frac{(12.99)^2}{2 \times 9.81} \).

Step 6: Calculate

Perform the calculation: \( h_{max} = \frac{168.74}{19.62} \approx 8.60 \, \text{m} \).

Key Concepts

Projectile Motion

Projectile motion refers to the path that an object follows when it is thrown or propelled near the surface of the Earth, and it is under the influence of gravity only. When dealing with projectile motion, two components are essential: the horizontal and the vertical. These components help determine the trajectory of the object.

• Horizontal motion is constant because there are no horizontal forces acting after the initial throw (assuming no air resistance).
• Vertical motion is influenced by gravity, causing the object to accelerate downward.

The combination of these motions results in a parabolic trajectory. Understanding projectile motion is crucial for solving problems that involve objects being launched or thrown at various angles.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is represented by the formula \[ KE = \frac{1}{2} mv^2 \]where \( m \) is the mass of the object and \( v \) is its velocity. For a projectile, kinetic energy helps in understanding the object's state while in motion. When the ball is launched upward, it possesses kinetic energy due to its velocity.

• This kinetic energy will be converted into potential energy as it reaches its highest point.
• At the highest point, the vertical component of kinetic energy becomes zero, indicating that all the kinetic energy has been converted into potential energy.

This conversion forms the basis for applying the conservation of energy principle to determine key parameters like the greatest height reached by the projectile.

Potential Energy

Potential energy, in the context of projectile motion, refers to the energy stored due to an object's position relative to a reference point, typically the ground. It is given by the formula \[ PE = mgh \]where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground. As a projectile rises, it gains potential energy while losing kinetic energy.

• Maximum potential energy is achieved at the highest point of the motion where kinetic energy from vertical motion is fully converted.
• The relationship between kinetic and potential energy forms the core of energy conservation principles in physics.

Understanding potential energy is critical for calculating the maximum height reached by a projectile using the energy conservation method.

Trigonometry in Physics

Trigonometry plays a vital role in physics, particularly when dealing with projectile motion. By using trigonometric functions, one can resolve the initial velocity into its horizontal and vertical components. For a projectile launched at an angle \( \theta \), the initial velocity \( v_i \) can be split as follows:

• Horizontal component: \( v_{ix} = v_i \cos(\theta) \)
• Vertical component: \( v_{iy} = v_i \sin(\theta) \)

In our given problem, by utilizing trigonometric functions, we are able to focus on the vertical component of the velocity to apply the energy conservation principles effectively.
Using these relations simplifies the process of analyzing projectile motion and allows for precise predictions about the path and duration of an object's flight.