Question

Therefore, for x(t) = 6 sin(t), the velocity at time t is given by v(t) = x'(t) = _______ and the acceleration at time t is given by _______?

Answer 1

The velocity of the function x(t) = 6 sin(t) is v(t) = 6 cos(t), and the acceleration is a(t) = -6 sin(t).

Step-by-step explanation:

For the function x(t) = 6 sin(t), we find the velocity as a function of time by taking the derivative with respect to time, v(t) = x'(t).

The derivative of 6 sin(t) with respect to time is 6 cos(t), therefore the velocity function is v(t) = 6 cos(t). To obtain the acceleration, we take the derivative of the velocity function with respect to time, a(t) = v'(t).

The derivative of 6 cos(t) is -6 sin(t), so the acceleration function is a(t) = -6 sin(t), which is the same as the original displacement function but with a negative sign.