Question

ann, a gymnast, weighs 40 kg. she is jumping on a trampoline that has a spring constant value of 176,400 \\(\frac{n}{m}\\). if she compresses the trampoline 20 cm, how high should she reach? \\(\boxed{}\\) meters

Explanation:

Step1: Identify the principle (Conservation of Energy)

We use the conservation of mechanical energy. The elastic potential energy stored in the trampoline (spring) will convert to gravitational potential energy at the maximum height. The formula for elastic potential energy is \( U_{s}=\frac{1}{2}kx^{2} \) and for gravitational potential energy is \( U_{g}=mgh \). At maximum height, \( U_{s}=U_{g} \), so \( \frac{1}{2}kx^{2}=mgh \).

Step2: Convert units

First, convert the compression distance \( x \) from centimeters to meters. \( 20\ cm = 0.2\ m \). The mass \( m = 40\ kg \), spring constant \( k = 176400\ \frac{N}{m} \), and \( g = 9.8\ \frac{m}{s^{2}} \).

Step3: Solve for height \( h \)

From \( \frac{1}{2}kx^{2}=mgh \), we can rearrange to solve for \( h \):

$$ h=\frac{kx^{2}}{2mg} $$

Substitute the values: \( k = 176400 \), \( x = 0.2 \), \( m = 40 \), \( g = 9.8 \)

$$ h=\frac{176400\times(0.2)^{2}}{2\times40\times9.8} $$

First, calculate the numerator: \( 176400\times0.04 = 7056 \)
Then, calculate the denominator: \( 2\times40\times9.8 = 784 \)
Now, divide: \( h=\frac{7056}{784}=9 \)

Answer:

\( 9 \)