Alright, let’s break down the problem into a series of steps, focusing on understanding the energy transformations for Jessica on the swing.
1. Given Information:
- Jessica’s mass ([tex]\(m\)[/tex]) = 44 kg
- Gravitational acceleration ([tex]\(g\)[/tex]) = 9.8 m/s[tex]\(^2\)[/tex]
- Height ([tex]\(h\)[/tex]) from which the swing is released = 3 meters
- Maximum velocity ([tex]\(v_{max}\)[/tex]) of the swing = 5 meters/second
2. Potential Energy Calculation:
Potential energy is given by the formula:
[tex]\[
PE = m \times g \times h
\][/tex]
Substituting the given values:
[tex]\[
PE = 44 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 3 \, \text{m}
\][/tex]
[tex]\[
PE = 1293.6000000000001 \, \text{Joules}
\][/tex]
So, the potential energy when the swing is at the maximum height of 3 meters is 1293.6 Joules.
3. Kinetic Energy at Maximum Velocity:
The kinetic energy is calculated using the formula:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
For maximum velocity [tex]\(v_{max} = 5 \, \text{m/s}\)[/tex]:
[tex]\[
KE_{\text{max}} = \frac{1}{2} \times 44 \, \text{kg} \times (5 \, \text{m/s})^2
\][/tex]
[tex]\[
KE_{\text{max}} = \frac{1}{2} \times 44 \times 25
\][/tex]
[tex]\[
KE_{\text{max}} = 22 \times 25
\][/tex]
[tex]\[
KE_{\text{max}} = 550 \, \text{Joules}
\][/tex]
So, the kinetic energy at the maximum velocity is 550 Joules.
4. Kinetic Energy at Zero Velocity (Initial State):
When the swing is first released from rest (at the highest point), the velocity is zero. Hence the kinetic energy is:
[tex]\[
KE_{\text{initial}} = 0 \, \text{Joules}
\][/tex]
Therefore, the energy Jessica has at different positions on the swing can be summarized as:
- At the highest point (3 meters):
- Potential Energy [tex]\( = 1293.6 \, \text{Joules} \)[/tex]
- Kinetic Energy [tex]\( = 0 \, \text{Joules} \)[/tex]
- At the lowest point, where the swing has maximum velocity (5 m/s):
- Potential Energy [tex]\( = 0 \, \text{Joules} \)[/tex] (since [tex]\(h = 0\)[/tex] at the bottom)
- Kinetic Energy [tex]\( = 550 \, \text{Joules} \)[/tex]
These values validate the energy conversions for Jessica on the swing, considering no energy is lost to friction or air resistance.
