Question

A large air-filled 0.100 -kg plastic ball is thrown up into the air with an initial speed of \(10.0 \mathrm{~m} / \mathrm{s}\). At a height of \(3.00 \mathrm{~m},\) the ball's speed is \(3.00 \mathrm{~m} / \mathrm{s}\). What fraction of its original energy has been lost to air friction?

Short answer

Solution: Step 1: Calculate the initial mechanical energy of the ball \(KE_{\text{initial}} = \frac{1}{2} × 0.100\,\text{kg} × (10.0\,\text{m/s})^2 = 5.0\,\text{J}\) \(E_{\text{initial}} = KE_{\text{initial}} = 5.0\,\text{J}\) Step 2: Calculate the final mechanical energy of the ball \(KE_{\text{final}} = \frac{1}{2} × 0.100\,\text{kg} × (3.00\,\text{m/s})^2 = 0.45\,\text{J}\) \(PE_{\text{final}} = 0.100\,\text{kg} × 9.81\,\text{m/s}^2 × 3.00\,\text{m} = 2.943\,\text{J}\) \(E_{\text{final}} = KE_{\text{final}} + PE_{\text{final}} = 0.45\,\text{J} + 2.943\,\text{J} = 3.393\,\text{J}\) Step 3: Calculate the fraction of energy lost to air friction \(E_{\text{lost}} = E_{\text{initial}} - E_{\text{final}} = 5.0\,\text{J} - 3.393\,\text{J} = 1.607\,\text{J}\) \(Fraction = \frac{E_{\text{lost}}}{E_{\text{initial}}} = \frac{1.607\,\text{J}}{5.0\,\text{J}} = 0.3214\) Answer: The fraction of energy lost due to air friction is 0.3214, or about 32.14%.

Step-by-step solution

Calculate the initial mechanical energy of the ball

To calculate the initial mechanical energy (E_initial) of the ball, we need to consider its kinetic energy and potential energy. The kinetic energy (KE) is given by the formula \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass of the ball and \(v\) is its initial speed. Given that the ball is thrown with an initial speed of 10 m/s, and its mass is 0.100 kg, the initial kinetic energy is: \(KE = \frac{1}{2} × 0.100\,\text{kg} × (10.0\,\text{m/s})^2\) The potential energy (PE) can be calculated using the formula \(PE = mgh\), where \(g\) is the acceleration due to gravity (approximately \(9.81\,\text{m/s}^2\)) and \(h\) is the initial height of the ball. Since the ball is thrown from height 0, the potential energy (PE_initial) is 0. Therefore, the initial mechanical energy (E_initial) is equal to the initial kinetic energy: \(E_{\text{initial}} = KE_{\text{initial}} + PE_{\text{initial}} = KE_{\text{initial}}\)

Calculate the final mechanical energy of the ball

The final mechanical energy (E_final) of the ball consists of its kinetic energy (KE_final) and potential energy (PE_final) at a height of 3.00 m and a speed of 3.00 m/s. First, calculate the final kinetic energy using the formula \(KE = \frac{1}{2}mv^2\): \(KE_{\text{final}} = \frac{1}{2} × 0.100\,\text{kg} × (3.00\,\text{m/s})^2\) Next, calculate the final potential energy using the formula \(PE = mgh\): \(PE_{\text{final}} = 0.100\,\text{kg} × 9.81\,\text{m/s}^2 × 3.00\,\text{m}\) Finally, find the total final mechanical energy (E_final) by summing the final kinetic energy and the final potential energy: \(E_{\text{final}} = KE_{\text{final}} + PE_{\text{final}}\)

Calculate the fraction of energy lost to air friction

Now that we have the initial and final mechanical energies, we can find the energy lost (E_lost) to air friction by: \(E_{\text{lost}} = E_{\text{initial}} - E_{\text{final}}\) Then, we can calculate the fraction of energy lost to air friction by dividing the energy lost by the initial energy: \(Fraction = \frac{E_{\text{lost}}}{E_{\text{initial}}}\) Calculate this fraction using the values obtained in Steps 1 and 2.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is quantified by the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \(m\) represents the mass of the object and \(v\) is its velocity. In our scenario, a ball thrown into the air starts with a certain amount of kinetic energy based on its speed. As it rises, part of this energy is converted to potential energy. If the ball reaches a state of rest momentarily at the peak of its trajectory, all of the initial kinetic energy would at that instant be converted into potential energy. However, due to the presence of air friction, the ball never reaches that state perfectly, and some of the kinetic energy is dissipated as heat and sound, which are not recoverable in a practical sense.

Potential Energy

Potential energy, on the other hand, is the stored energy in an object due to its position relative to a force, usually gravitational force for objects near the Earth's surface. The formula for gravitational potential energy is:
\[ PE = mgh \]
where \(g\) is the acceleration due to gravity and \(h\) is the height above a reference point. In the exercise, the ball, when thrown upwards, gains potential energy as it rises because it is moving against the gravitational force. This type of energy is at its maximum when the ball reaches the highest point of its path. Again, because of air friction, the actual maximum height achieved is lower than it would be in a vacuum, indicating a loss of mechanical energy.

Air Friction

Air friction, also known as air resistance, refers to the forces that are opposed to the motion of an object through the air. It is a form of fluid resistance, where fluid in this context refers to gases and liquids. As the ball travels through the air, it collides with air molecules, which creates a force that opposes its motion. This force does work on the ball, and thereby reduces the ball's mechanical energy. In the calculation of energy loss due to air friction, we are concerned with how much energy is not available to be converted between kinetic and potential energy because it has been transformed into other forms of energy, such as thermal energy due to the friction between the ball and the air.

Energy Conservation

The principle of energy conservation states that in a closed system, energy cannot be created or destroyed, only transformed from one form to another. When there is no air resistance (in a vacuum), the mechanical energy (sum of kinetic and potential energy) of a system would remain constant, implying that the energy lost from kinetic energy is exactly gained by potential energy, or vice versa. However, in real-world scenarios where air friction is present, some mechanical energy of the object is always lost to this frictional force. This transformed energy is generally not usable mechanical energy, which leads to what we perceive as a 'loss' of energy in terms of useful mechanical output. Calculating the fraction of energy lost to air friction involves understanding how much mechanical energy was initially present and how much remains after the ball has moved through the air.