Problem 76 - College Physics

Question

A \(1.20-\mathrm{kg}\) ball is projected straight upward with an initial speed of \(18.5 \mathrm{~m} / \mathrm{s}\) and reaches a maximum height of \(14.7 \mathrm{~m} .\) (a) Show numerically that total mechanical energy is not conserved during this part of the ball's motion. (b) Determine the work done on the ball by the force of air resistance. (c) Calculate the average air resistance force on the ball and the ball's average acceleration.

Step-by-Step Solution

Answer

Total mechanical energy is not conserved. The work done by air resistance is approximately 32.77 J. Average air resistance is about 2.23 N, with average acceleration of 7.93 m/s².

Step 1: Calculate Initial Kinetic Energy

Use the formula for kinetic energy: \[ KE_i = \frac{1}{2}mv^2 \]where \( m = 1.20 \ kg \) and \( v = 18.5 \ m/s \). Plug in the values:\[ KE_i = \frac{1}{2} \times 1.20 \times (18.5)^2 = 205.35 \ J \]

Step 2: Calculate Final Potential Energy

At maximum height, all kinetic energy is converted to potential energy. Use the formula for potential energy:\[ PE_f = mgh \]where \( g = 9.8 \ m/s^2 \) and \( h = 14.7 \ m \). Plug in the values:\[ PE_f = 1.20 \times 9.8 \times 14.7 = 172.584 \ J \]

Step 3: Compare Total Mechanical Energies

Mechanical energy should be conserved if there are no non-conservative forces acting:\[ E_i = KE_i = 205.35 \ J \] and\[ E_f = PE_f = 172.584 \ J \].Since \( E_i eq E_f \), mechanical energy is not conserved.

Step 4: Calculate Work Done by Air Resistance

The work done by air resistance equals the change in mechanical energy:\[ W = E_i - E_f = 205.35 - 172.584 = 32.766 \ J \]

Step 5: Calculate Average Air Resistance Force

Use the work-energy principle, where work done is equal to force times distance:\[ W = F_{avg} \times d \]\[ F_{avg} = \frac{W}{d} = \frac{32.766}{14.7} \approx 2.23 \ N \]

Step 6: Calculate the Ball's Average Acceleration

Using Newton's second law, find acceleration: \[ F_{net} = ma \]The net force is the sum of gravitational force and air resistance:\[ F_{net} = F_{gravity} - F_{avg} = mg - F_{avg} \]\[ a = \frac{F_{net}}{m} = \frac{mg - F_{avg}}{m} = g - \frac{F_{avg}}{m} \]\[ a = 9.8 - \frac{2.23}{1.20} \approx 7.93 \ m/s^2 \]

Key Concepts

Kinetic Energy

Kinetic energy is a form of mechanical energy that an object possesses due to its motion. It is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) represents the mass of the object and \( v \) its velocity. Kinetic energy depends directly on both the mass of the object and the square of its velocity. This means that even a small increase in speed can lead to a significant increase in kinetic energy.

Understanding kinetic energy is crucial for analyzing movements and forces acting on objects. For instance, when the ball in the exercise is thrown upwards with an initial speed of \( 18.5 \, \mathrm{m/s} \), it starts with a specific amount of kinetic energy, calculated to be approximately \( 205.35 \, \mathrm{J} \). As the ball rises, kinetic energy is gradually converted into potential energy until it reaches its peak height.

Potential Energy

Potential energy is another important form of mechanical energy. It is the energy stored in an object due to its position in a force field, such as gravity. The potential energy related to height is calculated with the formula \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

In the given exercise, as the ball reaches the maximum height of \( 14.7 \, \mathrm{m} \), its velocity becomes zero for a moment, and all its initial kinetic energy has been converted to gravitational potential energy. At this point, the potential energy of the ball is \( 172.584 \, \mathrm{J} \). This transformation highlights the ongoing interchange between kinetic and potential energy as the object moves under the influence of gravity.

Conservation of Energy

The principle of conservation of energy states that energy in a closed system remains constant over time; it can neither be created nor destroyed but can change from one form to another. In an ideal scenario, without external forces, the total mechanical energy (the sum of kinetic and potential energies) would remain constant. However, in real-world events, like the ball's flight in the exercise, other forces such as air resistance come into play.

During the ball's upward motion, energy is not conserved perfectly. Initially, the mechanical energy is the kinetic energy of \( 205.35 \, \mathrm{J} \). But at the peak, only \( 172.584 \, \mathrm{J} \) is available as potential energy. This discrepancy is due to work done by non-conservative forces like air resistance. The missing energy is accounted for by the work done by air resistance, calculated as \( 32.766 \, \mathrm{J} \). Understanding these energy transformations is essential for solving practical physics problems.