Question

The position function of an object as a function of time is s(t)=t³-12 t+36), in metres at time, t, in seconds.

a) Determine the velocity and acceleration functions.
b) Determine acceleration when velocity is zero.
c) Find when the object is speeding up using the inequalities chart.

Answer 1

a) The velocity function is v(t) = 3t² - 12. b) The object has zero velocity at t = 2 and t = -2. c) The object is speeding up when t > 0.

Step-by-step explanation:

a) To determine the velocity function, we need to differentiate the position function with respect to time. Differentiating t³ gives us 3t² and differentiating -12t gives us -12. So, the velocity function is v(t) = 3t² - 12.

b) To find when the velocity is zero, we need to solve the equation 3t² - 12 = 0. Simplifying, we get t² - 4 = 0, which gives us t = ±2. So, the object has zero velocity at t = 2 and t = -2.

c) To determine when the object is speeding up, we need to look at the sign of the acceleration function. The acceleration function is the derivative of the velocity function, which is a(t) = 6t. The object is speeding up when the acceleration is positive, which occurs for t > 0.