Question

The position of a particle moving along the x-axis is given by x(t) = t³ - 12t² - 27t with t in [0, 2]. What is the velocity of the particle at time t?

a) 3t² - 24t - 27
b) 3t² + 24t - 27
c) 3t² - 24t + 27
d) 3t² + 24t + 27

Answer 1

The velocity of the particle at time t, given the position function x(t) = t³ - 12t² - 27t, is found by differentiating x(t) with respect to t, resulting in the velocity function v(t) = 3t² - 24t - 27.

Step-by-step explanation:

The velocity of a particle is found by taking the derivative of its position function with respect to time. Given the position function x(t) = t³ - 12t² - 27t, the velocity v(t) can be calculated as follows:

• Differentiate each term of the position function with respect to t.
• For t³, the derivative is 3t².
• For -12t², the derivative is -24t.
• And for -27t, the derivative is -27.

Combining these, the velocity function v(t) becomes 3t² - 24t - 27.

Therefore, the correct answer for the velocity of the particle at time t is option a) 3t² - 24t - 27.