Question

a metal ball attached to a spring is bouncing up and down. at 0.5 second, the spring is at its highest position, which is 40 centimeters above its midline, and at 1.5 seconds, the spring reaches its lowest position, which is 40 centimeters below its midline. which function could represent the ball’s height, h(t), as a function of time, t?
options:

• h(t) = sin(πt) + 1.5
• h(t) = sin(πt) + 40
• h(t) = 40sin(πt)
• h(t) = 1.5sin(πt)

Explanation:

Step1: Analyze Amplitude

The maximum height above midline and minimum below are both 40 cm, so amplitude \( A = 40 \).

Step2: Analyze Period

Time between max and min is \( 1.5 - 0.5 = 1 \) second, so period \( T = 2 \) seconds. The formula for period of \( \sin(Bt) \) is \( \frac{2\pi}{B} \), so \( \frac{2\pi}{B}=2 \Rightarrow B = \pi \).

Step3: Check Midline and Function Form

The motion is a sinusoidal function with amplitude 40, period \( 2 \) (so \( B = \pi \)), and midline at 0 (since max is +40, min is -40, midline is 0). The general form for such a function is \( h(t)=A\sin(Bt) \), here \( A = 40 \), \( B=\pi \), so \( h(t)=40\sin(\pi t) \).

Step4: Eliminate Other Options

• \( h(t)=\sin(\pi t)+1.5 \): Amplitude 1, midline 1.5, doesn't match 40 cm amplitude.
• \( h(t)=\sin(\pi t)+40 \): Amplitude 1, midline 40, max 41, min 39, not 40 above/below.
• \( h(t)=1.5\sin(\pi t) \): Amplitude 1.5, too small.

Answer:

\( h(t) = 40\sin(\pi t) \) (the third option)