Question

Suppose a particle's position is given by the function g(t)=(t²+1)³. Find an expression for the particle's acceleration.

Answer 1

To determine a particle's acceleration from its position function g(t)=(t²+1)³, we calculate the first and second derivatives of g(t), with the second derivative providing the acceleration. This results in the expression for acceleration: a(t) = 6(t²+1)² + 12t²(t²+1).

Step-by-step explanation:

To find the particle's acceleration when the position is given by the function g(t)=(t²+1)³, we start by finding the first and second derivatives of the position function with respect to time, as the first derivative represents velocity and the second derivative represents acceleration.

1. First, find the first derivative which is the velocity: v(t) = g'(t).

   v(t) = d/dt [(t²+1)³]

   v(t) = 3(t²+1)²(2t)

   Now, we have an expression for velocity, v(t) = 6t(t²+1)².

2. Next, find the second derivative which is the acceleration: a(t) = v'(t).

   a(t) = d/dt [6t(t²+1)²]

   Apply the product rule to find the derivative of the product of functions:

   a(t) = 6(t²+1)² + 12t²(t²+1)

This gives us an expression for the particle's acceleration, a(t).