Final answer:
The velocity of a particle at any time t is found by calculating the first derivative of the position function s(t) with respect to time. To determine the speed, the magnitude of the velocity vector is taken, using absolute values if the motion is one-dimensional.
Step-by-step explanation:
For a particle where s(t) represents the distance from the origin, the velocity of the particle at any time t is the first derivative of the position function with respect to time. If the position function of a particle was given, we could find the instantaneous velocity at particular times by evaluating the derivative at those times.
As an example, if the velocity v(t) was given by the equation v(t) = 3.0 + 1.5t² m/s, then we would substitute the values of t into this equation to find the instantaneous velocity at those specific times. For instance, to find the velocity at t = 2.0 s, you would compute v(2.0 s) = [3.0 + 1.5(2.0)²] m/s = 9.0 m/s.
The speed of the particle is the magnitude of the velocity vector, which is always a non-negative value. It would be calculated by taking the absolute value of the velocity if the motion is one-dimensional or by computing the magnitude of the velocity vector if the motion is in multiple dimensions.
The velocity of the particle at any time t can be calculated by taking the derivative of the distance function s(t). If s(t) is given in feet, then the units of velocity will be feet per second (ft/s).
For example, if s(t) = 3t^2 + 2t + 1, then the velocity function v(t) can be found by taking the derivative of s(t) with respect to t. So, v(t) = 6t + 2.
To find the instantaneous velocity at a specific time t, substitute the value of t into the velocity function. For example, to find the instantaneous velocity at t = 0.25 s, substitute 0.25 into v(t).