A Spring's Bounce

Part of the Graphs of Trig Functions section of Coddy's Trigonometry journey. Lesson 42 of 49.

A weight hanging on a spring bobs up and down about its resting point. Pull it 3 cm above rest and let go: it falls, swings 3 cm below, and comes back up. If one bounce takes 2 seconds, then ignoring friction, which slowly shrinks the bounce, its height above rest after t seconds is

$y = 3\cos\left(\pi\cdot t\right)$

This model uses radians: the angle must turn 2π in one period of 2 seconds, so b = 2π ÷ 2 = π. It starts at its top, so it is a cosine with a positive a. Released from below rest, it would start at its bottom: a negative a.

$The spring model over four seconds: a cosine wave of amplitude 3 and period 2$

Height above rest in cm against time in seconds.

Challenge

A weight is pulled 4 cm below its resting point and let go. It bounces once every second. Ignoring friction, write its height above rest as $y = a\cdot \cos\left(b\cdot t\right)$ , with b a multiple of π written like 3πt. Press Check.