Problem 39 - University Physics with Modern Physics

Question

A basketball of mass \(0.624 \mathrm{~kg}\) is shot from a vertical height of \(1.2 \mathrm{~m}\) and at a speed of \(20.0 \mathrm{~m} / \mathrm{s}\). After reaching its maximum height, the ball moves into the hoop on its downward path, at \(3.05 \mathrm{~m}\) above the ground. Using the principle of energy conservation, determine how fast the ball is moving just before it enters the hoop.

Step-by-Step Solution

Answer

Answer: The speed of the basketball just before it enters the hoop is approximately 19.1 m/s.

Step 1: Identifying the known parameters and unknowns

In this problem, we have the following known quantities: the mass of the basketball (m = 0.624 kg), its initial speed (v0 = 20.0 m/s), and the initial height (h0 = 1.2 m). We need to find the final speed of the ball (vf) when it is at a height of 3.05 m above the ground (hf = 3.05 m).

Step 2: Calculate the initial potential and kinetic energy

We first need to calculate the initial kinetic and potential energy of the ball. The kinetic energy (KE) formula is given by: KE = 0.5 * m * v0^2 Note that the gravitational potential energy (PE) formula is given by: PE = m * g * h where g is the acceleration due to gravity (approximately 9.81 m/s²). Now, calculate the initial kinetic energy (KE_initial) and initial potential energy (PE_initial): KE_initial = 0.5 * 0.624 kg * (20.0 m/s)² = 124.8 J PE_initial = 0.624 kg * 9.81 m/s² * 1.2 m = 7.33 J

Step 3: Calculate the final potential energy

Next, we need to calculate the final potential energy (PE_final) when the ball is at a height of 3.05 m: PE_final = 0.624 kg * 9.81 m/s² * 3.05 m = 18.32 J

Step 4: Apply the principle of energy conservation

Since energy is conserved, we have: initial total energy = final total energy KE_initial + PE_initial = KE_final + PE_final Substituting the known values: 124.8 J + 7.33 J = KE_final + 18.32 J

Step 5: Calculate the final kinetic energy

Now, calculate KE_final: KE_final = (124.8 + 7.33 - 18.32) J = 113.81 J

Step 6: Solve for the final speed of the ball

Recall that the kinetic energy formula is: KE = 0.5 * m * v^2 We already know KE_final. To find the final speed of the ball (vf), we need to rearrange the formula and solve for vf: vf² = (2 * KE_final) / m Now, substitute the values: vf² = (2 * 113.81 J) / 0.624 kg vf² = 364.1 Now, take the square root to find vf: vf = sqrt(364.1) = 19.1 m/s The speed of the basketball just before it enters the hoop is approximately 19.1 m/s.

Key Concepts

Kinetic Energy

Understanding kinetic energy is crucial in solving physics problems involving moving objects. Kinetic energy (\textbf{KE}), often visualized as the energy of motion, is quantified by the equation \[ KE = \frac{1}{2} m v^2 \]
In this formula, \(m\) represents the mass of the object and \(v\) is its velocity. An object gains kinetic energy as it speeds up and loses it when slowing down. For instance, in the basketball problem, we initially gathered the kinetic energy using its mass and initial speed, depicting how much energy is tied up in its movement.

Potential Energy

Potential energy (\textbf{PE}) relates to the position or condition of an object. Gravitational potential energy is the most relevant form for objects near Earth's surface, calculated through the equation \[ PE = m g h \]
Where \(g\) is the acceleration due to gravity (\(9.81 \text{m/s}^2\)), and \(h\) is the object's height above a reference point. The basketball's potential energy at different heights showcases this object's stored energy due to gravity, convertible to kinetic energy when the basketball falls.

Conservation of Energy

The principle of conservation of energy states that within a closed system, energy can neither be created nor destroyed, only transformed from one form to another. This key principle allows us to set up an equation in our basketball problem that equates the sum of kinetic and potential energy at one point with that sum at another point. Thus, by knowing the initial total energy, we can deduce the final kinetic energy as the basketball falls toward the hoop. The formula we used, \[ KE_{\text{initial}} + PE_{\text{initial}} = KE_{\text{final}} + PE_{\text{final}} \],
effectively captures the essence of energy conservation in the context of the problem.

Kinematics in Physics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces causing such motion. It encompasses concepts like velocity, acceleration, and the physics of projectiles. In addressing our basketball problem, kinematics intertwines with energy principles. For example, the ball’s height at any point during its journey affects both its potential energy and its kinetic energy, demonstrating a kinematic relationship between position and velocity. Solving for the basketball's final speed required us to piece this relationship together—and in doing so, we practiced the applications of kinematics in the realm of energy conservation.