Problem 17 - University Physics with Modern Physics

Question

Adult cheetahs, the fastest of the great cats, have a mass of about 70 kg and have been clocked to run at up to 72 mi/h (32 m/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) 35,840 Joules, (b) factor of 4.

Step 1: Understanding Kinetic Energy

Kinetic energy (KE) is the energy that an object possesses due to its motion. The formula to calculate the kinetic energy of an object is given by: \[ KE = \frac{1}{2} m v^2 \]where \( m \) is the mass of the object in kilograms (kg), and \( v \) is its velocity in meters per second (m/s).

Step 2: Calculating Initial Kinetic Energy

First, let's substitute the mass (\( m = 70 \) kg) and velocity (\( v = 32 \) m/s) of the cheetah into the kinetic energy formula:\[ KE = \frac{1}{2} \times 70 \times (32)^2 \]Calculate \( v^2 \):\[ 32^2 = 1024 \]Now, substitute back into the formula:\[ KE = \frac{1}{2} \times 70 \times 1024 = 35 \times 1024 = 35840 \text{ Joules} \]

Step 3: Understanding How Kinetic Energy Changes with Speed

The kinetic energy depends on the square of the velocity. So, if the velocity is doubled, the kinetic energy changes according to the square of the factor by which the velocity is increased.

Step 4: Doubling the Velocity

To see how the kinetic energy changes when the velocity is doubled, substitute the new velocity, which is \( 2v = 2 \times 32 = 64 \) m/s, into the kinetic energy formula:\[ KE_{new} = \frac{1}{2} m (2v)^2 \]\[ KE_{new} = \frac{1}{2} \times 70 \times (64)^2 \]Calculate \( (64)^2 \):\[ 64^2 = 4096 \]So,\[ KE_{new} = \frac{1}{2} \times 70 \times 4096 = 35 \times 4096 = 143360 \text{ Joules} \]

Step 5: Finding the Factor of Change in Kinetic Energy

To find the factor by which the kinetic energy increases, divide the new kinetic energy by the initial kinetic energy:\[ \text{Factor} = \frac{KE_{new}}{KE} = \frac{143360}{35840} \]Calculate the factor:\[ \text{Factor} = 4 \]

Key Concepts

Velocity

Velocity describes how fast something is moving and specifies its direction. It is essential for calculating kinetic energy because it's one of the main variables in the kinetic energy formula. Velocity is measured in meters per second (m/s), indicating how many meters an object travels per second in a specified direction.
When discussing velocity, it is important to remember:

• Velocity is a vector quantity, meaning it has both magnitude (speed) and direction.
• In the kinetic energy equation, velocity is squared. This means that even small changes in velocity result in significant changes in kinetic energy.
• For the cheetah running at its full speed, its velocity was 32 m/s.

Understanding these basics of velocity helps to grasp why doubling the velocity quadruples the kinetic energy, as kinetic energy depends on velocity raised to the power of two, making it highly sensitive to changes in speed.

Mass

Mass is a measure of the amount of matter in an object, measured in kilograms (kg). It is another key factor in the calculation of kinetic energy, as indicated by its presence in the kinetic energy formula: \[ KE = \frac{1}{2} m v^2 \]Here, mass is represented as \( m \).
The significance of mass in kinetic energy is highlighted by:

• Mass is a scalar quantity, meaning it only has magnitude and no direction.
• In the kinetic energy formula, mass is directly proportional to kinetic energy. Therefore, doubling the mass doubles the kinetic energy, assuming velocity stays the same.
• For our cheetah example, the mass was given as 70 kg, showing us that heavier objects at the same speed have more kinetic energy.

Considering mass helps us comprehend how different animals or objects with varying weights have different kinetic energies even when moving at the same velocity.

Energy Transformation

Energy transformation is the process by which energy changes from one form to another. In the context of a cheetah running, chemical energy stored in its muscles is transformed into kinetic energy as it accelerates.
Energy transformations are crucial for several reasons:

• Kinetic energy is the energy of motion, derived from the chemical energy in food converted by the body.
• As speed increases, the transformation results in increased kinetic energy, highlighting the efficiency of energy conversion in animals.
• If velocity doubles, as explored in the exercise, the kinetic energy increases fourfold, showcasing how energy transformations can exponentially increase with speed changes due to the squared relationship in the kinetic energy formula.

This concept helps in understanding not only how animals use and transform energy to move, but also illustrates basic principles of physics where energy is always conserved, just changed from one form to another in dynamic processes.