Harmonic Motion Simulation: Equations of Displacement, Velocity, and Acceleration

What are the equations of position, velocity, and acceleration in a harmonic motion simulation?

Given the data of an amplitude of 3 cm and a time of 1.5 s, how can we derive the equations for displacement (x(t)), velocity (v(t)), and acceleration (a(t))?

Equations of Displacement, Velocity, and Acceleration

Using the given data of an amplitude of 3 cm and a time of 1.5 s, we have derived the equations for displacement (x(t)), velocity (v(t)), and acceleration (a(t)).

To develop the equations of position, velocity, and acceleration for the given simulation with an amplitude of 3 cm and a time of 1.5 s, we will follow the steps outlined below:

Position Equation (x(t)):

Given the amplitude (A) of 3 cm, we can define the displacement (x) as a function of time (t) using the equation for simple harmonic motion:

x(t) = A * cos(2πt/T)

Substituting the values, we have:

x(t) = 3 * cos((2π * 1.5)/T)

Velocity Equation (v(t)):

To find the velocity (v), we need to differentiate the position equation with respect to time (t):

v(t) = dx(t)/dt

Differentiating the position equation gives us:

v(t) = -(2πA/T) * sin(2πt/T)

Substituting the values, we have:

v(t) = -(2π * 3/T) * sin((2π * 1.5)/T)

Acceleration Equation (a(t)):

To find the acceleration (a), we differentiate the velocity equation with respect to time (t):

a(t) = dv(t)/dt

Differentiating the velocity equation gives us:

a(t) = -(4π²A/T²) * cos(2πt/T)

Substituting the values, we have:

a(t) = -(4π² * 3/T²) * cos((2π * 1.5)/T)