Jessica is visiting a park with her mother. Jessica sits on a swing. Her mother pulls the swing to a height of 3 meters above the ground and lets it go. The image shows Jessica at three positions on the swing. Jessica's mass is 44 kilograms and the maximum velocity of the swing is 5 meters/second. What's the energy she has at each position shown? Ignore friction and air resistance. Use [tex]\(g=9.8 \, \text{m/s}^2\)[/tex], [tex]\(PE = m \times g \times h\)[/tex], and [tex]\(KE = \frac{1}{2} m v^2\)[/tex].

Given:
- [tex]\(KE = 0\)[/tex] joules
- [tex]\(KE = 550\)[/tex] joules
- [tex]\(PE = 862.4\)[/tex] joules



Answer :

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.