In simple harmonic motion (SHM), the velocity of an object can be derived from its displacement, amplitude, and angular speed. The equation for the displacement of an object undergoing SHM is given by:
x(t) = A * cos(ωt)
Where: x(t) is the displacement of the object at time t A is the amplitude of the motion ω is the angular speed or angular frequency, given by ω = 2πf, where f is the frequency of the motion.
To find the velocity, we can differentiate the displacement equation with respect to time:
v(t) = d/dt (A * cos(ωt))
Using the chain rule, the derivative of cos(ωt) with respect to ωt is -ω * sin(ωt). The derivative of A * cos(ωt) with respect to ωt is A * (-ω * sin(ωt)).
Therefore, the velocity equation for an object undergoing simple harmonic motion is:
v(t) = -A * ω * sin(ωt)
Alternatively, you can express the velocity equation in terms of displacement as:
v(x) = ± ω * √(A² - x²)
Where: v(x) is the velocity of the object at displacement x from the equilibrium position. ± indicates that the velocity can be positive or negative, depending on the direction of motion. ω is the angular speed or angular frequency. A is the amplitude of the motion.
This equation shows that the velocity is maximum (|v(x)| = ω * A) when the displacement is zero, and it becomes zero when the displacement is equal to the amplitude (x = ± A).