The velocity of an object undergoing simple harmonic motion can be derived using the equation for displacement as a function of time. The general equation for displacement in simple harmonic motion is given by:
x(t) = A * cos(ωt + φ)
where:
- x(t) is the displacement of the object at time t.
- A is the amplitude of the motion.
- ω is the angular speed (angular frequency) of the motion.
- φ is the phase constant.
To find the velocity, we need to differentiate the displacement equation with respect to time:
v(t) = dx(t)/dt
Differentiating x(t) with respect to t gives:
v(t) = -A * ω * sin(ωt + φ)
So, the derived expression for the velocity of an object in simple harmonic motion, in terms of displacement (x), amplitude (A), and angular speed (ω), is:
v(t) = -A * ω * sin(ωt + φ)
Note that the negative sign arises because the derivative of the cosine function (which represents displacement) is the negative sine function.