Problem 13 - University Physics with Modern Physics

Question

Animal Energy. BIO Adult cheetahs, the fastest of the great cats, have a mass of about 70 \(\mathrm{kg}\) and have been clocked running at up to 72 \(\mathrm{mph}(32 \mathrm{m} / \mathrm{s})\) . (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) The kinetic energy quadruples if speed is doubled.

Step 1: Write the formula for kinetic energy

The formula for kinetic energy \( KE \) of an object with mass \( m \) moving with velocity \( v \) is given by:\[KE = \frac{1}{2} m v^2\]This formula will be used to calculate the kinetic energy of the cheetah.

Step 2: Calculate the kinetic energy of the cheetah

Given that the mass \( m \) of the cheetah is \( 70 \, \text{kg} \) and its speed \( v \) is \( 32 \, \text{m/s} \), we substitute these values into the kinetic energy formula:\[KE = \frac{1}{2} \times 70 \, \text{kg} \times (32 \, \text{m/s})^2\]Calculate the expression:\[KE = \frac{1}{2} \times 70 \times 1024 = 35 \times 1024 = 35840 \, \text{Joules}\]

Step 3: Determine how kinetic energy changes with doubled speed

To find out how the kinetic energy changes if the speed is doubled, we start by letting the new speed be \( 2v \). Substitute this new velocity into the kinetic energy formula:\[KE_{new} = \frac{1}{2} m (2v)^2 = \frac{1}{2} m \times 4v^2\]This means the kinetic energy becomes:\[KE_{new} = 4 \times \left(\frac{1}{2} m v^2\right) = 4 \times KE\]Thus, the cheetah's kinetic energy is multiplied by 4 when its speed is doubled.

Key Concepts

Physics Problems

Solving physics problems involves understanding the principles and laws that govern the physical world. In the case of kinetic energy, it’s important to start with identifying the given data points, such as the mass and velocity of an object, and to clearly understand what the problem asks you to find.

• First, read the problem carefully to extract all the necessary details. Here, the mass of the cheetah is 70 kg and its speed is 32 m/s.
• Then, determine what you need to calculate—in this instance, the kinetic energy at this speed and how it changes with speed variations.

Having a systematic approach to physics problems can make the process more manageable and less daunting. By breaking down a problem into smaller, sequential steps, you can tackle even complex questions with ease.

Energy Calculation

Energy calculations, particularly for kinetic energy, involve applying the known formula to the problem at hand. Let's delve into calculating the kinetic energy of the cheetah. The formula for kinetic energy is expressed as:\[ KE = \frac{1}{2} m v^2 \]

• Insert the provided values: the mass \( m = 70 \, \text{kg} \) and the velocity \( v = 32 \, \text{m/s} \).
• Perform the calculation as follows: \[ KE = \frac{1}{2} \times 70 \, \text{kg} \times (32 \, \text{m/s})^2 \]
• First calculate the square of the velocity: \((32)^2 = 1024\).
• Then multiply by the mass and the constant: \(35 \times 1024 = 35840 \, \text{Joules} \).

This step confirms the amount of kinetic energy the cheetah possesses at a speed of 32 m/s. Properly understanding these calculations strengthens your grasp on how kinetic energy varies with mass and velocity.

Formula Application

Applying formulas correctly is crucial in physics when exploring relationships such as between velocity and kinetic energy. In the exercise, you explored how changing the cheetah's speed influences its kinetic energy.

• When speed is doubled, assume the new speed is \(2v\). Substitute into the formula: \[ KE_{new} = \frac{1}{2} m (2v)^2 = \frac{1}{2} m \times 4v^2 \]
• This transformation shows that the kinetic energy is multiplied by a factor of four.
• Using the original kinetic energy value, you can further demonstrate this increase: \[ KE_{new} = 4 \times KE \]

Understanding the relationship between speed and kinetic energy through formulas deepens your comprehension of how physical changes impact energy, preparing you for more intricate physics challenges.