Question

practice: kinetic and potential energy #2
answer the following questions. make sure to show all work to receive credit. you may need a separate sheet of paper.

1. what is the energy of a 2.0 kg plant that is on a windowsill 1.3 m high?
2. what is the energy of a 7.18 kg bowling ball that is rolling at a speed of 8.5 m/s?
3. an apple has 1.06 j of energy. the apple sits on a counter 0.914 m high. what is the mass of the apple?
4. two objects are being lifted by a machine. one object has a mass of 2 kg, and is lifted at a speed of 2 m/s. the other has a mass of 4 kg and is lifted at a rate of 3 m/s.

a. which object has more kinetic energy while it is being lifted?
b. what does this tell you about the variable(s) that impact kinetic energy?
c. which object had more potential energy when it was lifted a distance of 10 m?
d. what does this tell you about the variable(s) that impact gravitational potential energy?

Explanation:

Step1: Identify the formula for gravitational potential energy

The formula for gravitational potential energy \( GPE = mgh \), where \( m \) is mass, \( g = 9.8\ m/s^{2} \) (acceleration due to gravity), and \( h \) is height.
For the plant: \( m = 2.0\ kg \), \( h = 1.3\ m \)
\( GPE=2.0\times9.8\times1.3 \)

Step2: Calculate the value

\( GPE = 2\times9.8\times1.3=25.48\ J \)

Step3: Identify the formula for kinetic energy

The formula for kinetic energy \( KE=\frac{1}{2}mv^{2} \), where \( m \) is mass and \( v \) is velocity.
For the bowling - ball: \( m = 7.18\ kg \), \( v = 8.5\ m/s \)
\( KE=\frac{1}{2}\times7.18\times(8.5)^{2} \)

Step4: Calculate the value

First, \( (8.5)^{2}=72.25 \)
Then, \( \frac{1}{2}\times7.18\times72.25 = 3.59\times72.25=259.3775\ J \)

Step5: Rearrange the gravitational potential energy formula for mass

From \( GPE = mgh \), we can solve for \( m=\frac{GPE}{gh} \)
Given \( GPE = 1.06\ J \), \( h = 0.914\ m \), \( g = 9.8\ m/s^{2} \)
\( m=\frac{1.06}{9.8\times0.914} \)

Step6: Calculate the value

\( 9.8\times0.914 = 8.9572 \)
\( m=\frac{1.06}{8.9572}\approx0.118\ kg \)

Step7: Calculate kinetic energy for the two objects

For object 1: \( m_1 = 2\ kg \), \( v_1 = 2\ m/s \)
\( KE_1=\frac{1}{2}\times2\times(2)^{2}=4\ J \)
For object 2: \( m_2 = 4\ kg \), \( v_2 = 3\ m/s \)
\( KE_2=\frac{1}{2}\times4\times(3)^{2}=2\times9 = 18\ J \)

Step8: Compare kinetic energies

Since \( 18>4 \), the \( 4 - kg \) object has more kinetic energy.

Step9: Calculate gravitational potential energy for the two objects (when \( h = 10\ m \))

For object 1: \( GPE_1=m_1gh_1=2\times9.8\times10 = 196\ J \)
For object 2: \( GPE_2=m_2gh_2=4\times9.8\times10=392\ J \)

Step10: Compare gravitational potential energies

Since \( 392>196 \), the \( 4 - kg \) object has more gravitational potential energy.

Answer:

1. The energy of the plant is \( 25.48\ J \)
2. The energy of the bowling - ball is \( 259.38\ J \)
3. The mass of the apple is approximately \( 0.12\ kg \)

4.
a. The \( 4 - kg \) object has more kinetic energy.
b. Kinetic energy depends on both mass and velocity (\( KE=\frac{1}{2}mv^{2} \)). A larger mass or a larger velocity (or both) will result in a larger kinetic energy.
c. The \( 4 - kg \) object has more potential energy.
d. Gravitational potential energy depends on mass and height (\( GPE = mgh \)). A larger mass or a larger height (or both) will result in a larger gravitational potential energy.