Question

at the top of a giant swing on the gymnastics high bar, candys velocity is \\(1\text{ m/s}\\), & she is \\(3.5\text{ m}\\) high. if her mass is \\(50\text{ kg}\\), what is her total mechanical energy at this instance? \\(te = ke + pe\\)

Explanation:

Identify given values and constants

We extract the physical parameters from the problem statement:

• Mass of the gymnast, \(m = 50\text{ kg}\)
• Velocity at the top, \(v = 1\text{ m/s}\)
• Height above the reference level, \(h = 3.5\text{ m}\)
• Acceleration due to gravity, \(g \approx 9.8\text{ m/s}^2\) (or \(9.81\text{ m/s}^2\))

Calculate kinetic energy

Kinetic energy represents energy of motion:

$$ KE = \frac{1}{2}mv^2 $$

Substituting the values:

$$ KE = \frac{1}{2} \times 50\text{ kg} \times (1\text{ m/s})^2 = 25\text{ J} $$

Calculate potential energy

Potential energy represents stored gravitational energy:

$$ PE = mgh $$

Substituting the values:

$$ PE = 50\text{ kg} \times 9.8\text{ m/s}^2 \times 3.5\text{ m} = 1715\text{ J} $$

(Note: If using \(g = 9.81\text{ m/s}^2\), \(PE = 1716.75\text{ J}\))

Sum to find total mechanical energy

Total mechanical energy is the sum of kinetic and potential energy:

$$ TE = KE + PE $$

Using \(g = 9.8\text{ m/s}^2\):

$$ TE = 25\text{ J} + 1715\text{ J} = 1740\text{ J} $$

Using \(g = 9.81\text{ m/s}^2\):

$$ TE = 25\text{ J} + 1716.75\text{ J} = 1741.75\text{ J} $$

Answer:

\(1740\text{ J}\) (or \(1741.75\text{ J}\) using \(g = 9.81\text{ m/s}^2\))