Problem 14 - College Physics

Question

\(\cdot\) \(\cdot\) Animal energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 \(\mathrm{kg}\) and have been clocked at up to 72 mph \((32 \mathrm{m} / \mathrm{s})\) . (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

Step-by-Step Solution

Answer

(a) 35840 J; (b) Factor of 4.

Step 1: Understanding Kinetic Energy Formula

Kinetic energy is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. We'll use this formula to calculate the kinetic energy of the cheetah.

Step 2: Calculate Kinetic Energy at Original Speed

First, we need to plug in the values for the cheetah's mass \( m = 70 \; \mathrm{kg} \) and speed \( v = 32 \; \mathrm{m/s} \) into the kinetic energy formula. \[KE = \frac{1}{2} \times 70 \times (32)^2 \]Calculate this to find the kinetic energy.

Step 3: Compute the Expression

Let's compute the expression: \[ KE = \frac{1}{2} \times 70 \times 1024 = 35 \times 1024 \]\[ KE = 35840 \; \mathrm{J} \]So, the kinetic energy of the cheetah is \( 35840 \; \mathrm{J} \).

Step 4: Calculate Kinetic Energy at Double Speed

To find the kinetic energy when the speed is doubled, first determine the new speed: New speed = \( 2 \times 32 = 64 \; \mathrm{m/s} \).Plug into the formula:\[KE_{new} = \frac{1}{2} \times 70 \times (64)^2 \]Calculate \( KE_{new} \).

Step 5: Compute Double Speed Expression

Compute the expression for the doubled speed:\[ KE_{new} = \frac{1}{2} \times 70 \times 4096 = 35 \times 4096 \]\[ KE_{new} = 143360 \; \mathrm{J} \]The kinetic energy at double speed is \( 143360 \; \mathrm{J} \).

Step 6: Determine the Factor of Change

Divide the new kinetic energy by the original kinetic energy to determine the factor of change:\[ \text{Factor} = \frac{143360}{35840} = 4 \]The kinetic energy increases by a factor of 4 when the speed is doubled.

Key Concepts

Kinetic Energy Formula

Understanding the Kinetic Energy Formula is key to grasping how moving objects store energy. The formula for kinetic energy (KE) is given by:

• \(KE = \frac{1}{2} m v^2 \)

Here, \( m \) represents the mass of the object, and \( v \) is its velocity. This formula reveals that kinetic energy depends on two factors: mass and speed. A slight increase in speed results in a greater increase in kinetic energy since velocity is squared in the formula.
Given that mass remains constant, doubling the speed increases the kinetic energy by a factor of four, illustrating the speed's significant impact on kinetic energy.

Physics Problem Solving

Solving physics problems involves systematically applying formulas to find unknown values. Start by understanding what is being asked, identifying the information given, and deciding on the appropriate formula to use. For the cheetah problem, we need to determine the kinetic energy at various speeds.
This process requires calculating kinetic energy at the original speed first. Then, you expand this process to solve for kinetic energy when the speed is doubled. Each step builds on the results of the previous one, which is a common strategy in physics problem solving.

Energy Transformation

Energy transformation refers to the change of energy from one form to another. In moving objects like a cheetah, chemical energy from food is transformed into kinetic energy. This transformation allows the animal to move swiftly.
When a cheetah accelerates, more chemical energy is turned into kinetic energy, illustrating the direct correlation between energy transformation and speed. Understanding this aids in grasping broader principles of energy conservation and applicability in the real world.

Speed and Velocity

Speed and velocity are crucial components in understanding how kinetic energy functions. Speed is the rate at which an object covers distance, while velocity includes both speed and direction. In the kinetic energy formula, it is the speed that is considered, but knowing the direction (velocity) is important for full comprehension in physics contexts.
For instance, in the given exercise, the cheetah reaches a speed of 32 m/s. If this speed is vectorized to include direction, it becomes velocity. However, since kinetic energy depends on the magnitude of speed, we focus purely on the scalar value for calculations.

Mathematical Calculations in Physics

Mathematical Calculations in Physics are central to understanding and predicting the behavior of physical systems. Using the kinetic energy formula involves straightforward multiplication and squaring of numbers. In our example with the cheetah, calculating kinetic energy requires multiplying mass by the square of velocity, then taking half of that product.
These calculations provide quantitative insight into energy transformations and help predict outcomes when variables change, such as doubling the speed and observing the effect on kinetic energy. Through consistent practice, students can develop proficiency in these calculations, which are foundational to deeper exploration in physics.