Tag Archives: comparing kinetic energy

College Physics by Openstax Chapter 7 Problem 10

(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.

Solution:

The translational kinetic energy of an object of mass m moving at speed v is KE=\frac{1}{2}mv^{2}.

Part A. The Velocity of the Elephant to have the same Kinetic Energy as the Sprinter

First, we need to solve for the kinetic energy of the sprinter. 

\begin{align*}
KE_{\text{sprinter}} & = \frac{1}{2} \left( 65.0\ \text{kg} \right)\left( 10.0\ \text{m}/\text{s} \right)^{2} \\
KE_{\text{sprinter}} & = 3250\ \text{J}
\end{align*}

Then, we need to equate this to the kinetic energy of the elephant with the velocity as the unknown. 

\begin{align*}
KE_{\text{elephant}} & = KE_{\text{sprinter}} \\
\frac{1}{2}\left( 3000\ \text{kg} \right) v^{2} & = 3250\ \text{J} \\
1500 v^{2} & = 3250 \\
v^{2} & = \frac{3250}{1500} \\
v & = \sqrt[]{\frac{3250}{1500}} \\
v & = 1.4720\ \text{m}/\text{s} \\
v & = 1.47\ \text{m}/\text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B. How Larger Energies Needed for the Movement of Larger Animals would Relate to Metabolic Rates

If the elephant and the sprinter accelerate to a final velocity of 10.0 m/s, then
the elephant would have a much larger kinetic energy than the sprinter.
Therefore, the elephant clearly has burned more energy and requires a faster
metabolic output to sustain that speed. \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

College Physics by Openstax Chapter 7 Problem 9

Compare the kinetic energy of a 20,000-kg truck moving at 110 km/h with that of an 80.0-kg astronaut in orbit moving at 27,500 km/h.

Solution:

The translational kinetic energy of an object of mass m moving at speed v is KE=\frac{1}{2}mv^{2}.

The Kinetic Energy of the Truck

For the truck, we are given the following:

\begin{align*}
m & = 20 000\ \text{kg} \\
v & = 110\ \frac{\text{km}}{\text{hr}} \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{hr}}{3600\ \text{s}} = 30.5556\ \text{m}/\text{s}
\end{align*}

Substitute these values to compute for the kinetic energy of the truck.

\begin{align*}
KE_{t} & = \frac{1}{2} mv^{2} \\
KE_{t} & = \frac{1}{2} \left( 20 000\ \text{kg} \right) \left( 30.5556\ \text{m}/\text{s} \right)^{2} \\
KE_{t} & = 9 336 446.9136\ \text{J} \\
KE_{t} & = 9.34 \times 10^{6} \ \text{J}
\end{align*}

The Kinetic Energy of the Astronaut

For the astronaut, we have the following given values

\begin{align*}
m & = 80\ \text{kg} \\
v & = 27 500\ \frac{\text{km}}{\text{hr}} \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{hr}}{3600\ \text{s}} = 7638.8889\ \text{m}/\text{s}
\end{align*}

The kinetic energy of the astronaut is calculated as

\begin{align*}
KE_{a} & = \frac{1}{2} mv^{2} \\
KE_{a} & = \frac{1}{2} \left( 80\ \text{kg} \right) \left( 7638.8889\ \text{m}/\text{s} \right)^{2} \\
KE_{a} & = 2 334 104 945 .0617\ \text{J} \\
KE_{a} & = 2.33 \times 10^{9} \ \text{J}
\end{align*}

Comparing the Kinetic Energies of the truck and the astronaut

\begin{align*}
\frac{KE_{a}}{KE_{t}} & = \frac{2 334 104 945 .0617\ \text{J}}{9 336 446.9136\ \text{J}} \\
\frac{KE_{a}}{KE_{t}} & = 250 \\
KE_{a} & = 250\ KE_{t} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The kinetic energy of the astronaut is 250 times larger than the kinetic energy of the truck.