Problem 56 - College Physics

Question

\(\bullet\) A fun-loving 11.4 \(\mathrm{kg}\) otter slides up a hill and then back down to the same place. If she starts up at 5.75 \(\mathrm{m} / \mathrm{s}\) and returns at 3.75 \(\mathrm{m} / \mathrm{s}\) , how much mechanical energy did she lose on the hill, and what happened to that energy?

Step-by-Step Solution

Answer

The otter lost 108.48 J of mechanical energy, converted mainly to thermal energy due to friction.

Step 1: Identify the Given Values

We're given that the otter has a mass of 11.4 kg. Her initial speed, when sliding up, is 5.75 m/s, and her final speed, when she returns to the starting point, is 3.75 m/s. Our task is to find out how much mechanical energy was lost due to the slide.

Step 2: Calculate Initial Kinetic Energy

The initial kinetic energy (KE_initial) when she starts sliding up can be calculated using the formula:\[KE_{initial} = \frac{1}{2} m v_{initial}^2\]Substituting the known values:\[KE_{initial} = \frac{1}{2} \times 11.4 \, \text{kg} \times (5.75 \, \text{m/s})^2 = 188.92 \, \text{Joules}\]

Step 3: Calculate Final Kinetic Energy

The final kinetic energy (KE_final) when she returns is calculated as follows:\[KE_{final} = \frac{1}{2} m v_{final}^2\]Substitute the given values:\[KE_{final} = \frac{1}{2} \times 11.4 \, \text{kg} \times (3.75 \, \text{m/s})^2 = 80.44 \, \text{Joules}\]

Step 4: Determine Mechanical Energy Lost

The mechanical energy lost is the difference between the initial and final kinetic energies:\[\text{Energy Lost} = KE_{initial} - KE_{final}\]Calculating:\[\text{Energy Lost} = 188.92 \, \text{Joules} - 80.44 \, \text{Joules} = 108.48 \, \text{Joules}\]

Step 5: Analyze What Happened to the Lost Energy

The lost mechanical energy, amounting to 108.48 Joules, was transformed into other forms of energy. Most likely, this includes thermal energy due to friction between the otter and the hill surface, and possibly sound energy.

Key Concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. It depends on two main factors: the object's mass and its velocity. The formula to calculate kinetic energy is:

• \[ KE = \frac{1}{2} m v^2 \]

Where \( KE \) is the kinetic energy, \( m \) is the mass, and \( v \) is the velocity of the object.
For instance, when the otter in the problem slides up the hill, she has an initial kinetic energy calculated using her speed of 5.75 m/s. Similarly, her kinetic energy can be recalculated when she returns to the bottom at a different speed of 3.75 m/s.
Observing these changes in energy is crucial to understanding how energy is conserved or transformed during motion.

Energy Transformation

Energy transformation is the process where energy changes from one form to another. Here, it's essential to understand that while the total energy of the system remains constant in an ideal situation (as per the principle of conservation of energy), the form of energy can change.
In the case of the otter sliding on the hill, her potential energy increases as she moves up the hill, and her kinetic energy decreases. Conversely, when sliding back down, potential energy decreases while kinetic energy increases.
However, in real-life scenarios, some of the kinetic energy might not fully convert back, leading to energy transformations into non-mechanical forms like thermal energy due to friction, which comes into play here.

Energy Loss Due to Friction

Energy loss due to friction is a common phenomenon where mechanical energy is converted into other forms like heat. Friction is a force that opposes motion between two surfaces in contact, causing some of the kinetic energy to dissipate.
In the otter's case, as she slides up and down the hill, friction between her and the surface of the hill converts a portion of her kinetic energy into heat. This is why, when she returns to the starting point, her speed is lower than when she began.

• The amount of energy loss can often be quantified, as in the exercise where the otter lost 108.48 Joules due to friction.

Understanding energy loss is important for grasping real-life applications of mechanical energy conservation and transformations.