Sure, let's solve each problem step-by-step.
### Problem 1:
A bicyclist is moving down a hill. Her position on the hill gives her [tex]$720 J$[/tex] of potential energy, and her movement gives her [tex]$680 J$[/tex] of kinetic energy. What is her total mechanical energy?
1. Potential Energy (PE): [tex]\( 720 \, J \)[/tex]
2. Kinetic Energy (KE): [tex]\( 680 \, J \)[/tex]
The total mechanical energy (TME) is the sum of the kinetic energy and the potential energy.
[tex]\[
\text{Total Mechanical Energy} = \text{Kinetic Energy} + \text{Potential Energy}
\][/tex]
[tex]\[
\text{Total Mechanical Energy} = 680 \, J + 720 \, J
\][/tex]
[tex]\[
\text{Total Mechanical Energy} = 1400 \, J
\][/tex]
So, the correct answer is:
B. [tex]$1400 J$[/tex]
### Problem 2:
A 3 kg ball is falling at 4 m/s, and is 6 meters off the ground. What is its total mechanical energy? (You will apply 3 formulas to solve this problem)
1. Mass (m): [tex]\( 3 \, kg \)[/tex]
2. Velocity (v): [tex]\( 4 \, m/s \)[/tex]
3. Height (h): [tex]\( 6 \, m \)[/tex]
4. Acceleration due to gravity (g): [tex]\( 9.8 \, m/s^2 \)[/tex]
To find the total mechanical energy, we need to calculate both the kinetic energy (KE) and the potential energy (PE) of the ball.
#### Step 1: Calculate Kinetic Energy (KE)
The formula for kinetic energy is:
[tex]\[
KE = 0.5 \times \text{mass} \times \text{velocity}^2
\][/tex]
[tex]\[
KE = 0.5 \times 3 \, kg \times (4 \, m/s)^2
\][/tex]
[tex]\[
KE = 0.5 \times 3 \times 16
\][/tex]
[tex]\[
KE = 0.5 \times 48
\][/tex]
[tex]\[
KE = 24 \, J
\][/tex]
#### Step 2: Calculate Potential Energy (PE)
The formula for potential energy is:
[tex]\[
PE = \text{mass} \times \text{gravity} \times \text{height}
\][/tex]
[tex]\[
PE = 3 \, kg \times 9.8 \, m/s^2 \times 6 \, m
\][/tex]
[tex]\[
PE = 3 \times 9.8 \times 6
\][/tex]
[tex]\[
PE = 176.4 \, J
\][/tex]
#### Step 3: Calculate Total Mechanical Energy (TME)
The total mechanical energy is the sum of the kinetic energy and the potential energy.
[tex]\[
TME = KE + PE
\][/tex]
[tex]\[
TME = 24 \, J + 176.4 \, J
\][/tex]
[tex]\[
TME = 200.4 \, J
\][/tex]
So, the correct answer is:
D. [tex]$200.4 J$[/tex]
