If an equation of motion of a particle is given by s(t)= Acos(ùt + ä), the particle is said to undergo simple harmonic motion. Find the velocity of the particle at time t. When is the velocity 0?

the velocity at time t is the derivative.

the derivative of s(t) with respect to t =
-A*sin(ùt + ä)(t) assuming that everything but t and s(t) are constent.
to find out when the velocity is 0 we just have to set the derivative = to 0 and solve for t so it looks like the velocity is 0 whent t=0 -A=0 or sin(u*t+a)=0

To find the velocity of the particle at any given time, we need to differentiate the equation of motion with respect to time (t).

Given: s(t) = Acos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

Differentiating s(t) with respect to t:
v(t) = -Aωsin(ωt + φ)

Now we have the equation for velocity at any time t.

To find when the velocity is zero, we need to solve the equation v(t) = 0.

0 = -Aωsin(ωt + φ)

Since sin(ωt + φ) can equal 0 at several points, we set the equation equal to zero and solve for ωt + φ.

-Aωsin(ωt + φ) = 0

sin(ωt + φ) = 0

To find when sin(ωt + φ) = 0, we look for the values of ωt + φ that satisfy this condition. The values of ωt + φ that satisfy this condition are:

ωt + φ = nπ, where n is an integer.

Now we solve for t:

ωt = nπ - φ

t = (nπ - φ) / ω

Thus, the velocity is zero at times t = (nπ - φ) / ω, where n is an integer.