Question

With distance measured in feet, a particle's position is given by r(t) = (1-2sin t, 1 - 2cos , 0). Find the speed at time t = 2 seconds.

Answer 1

To find the speed of a particle at time t = 2 seconds, differentiate the position function r(t) to obtain the velocity vector, then calculate the magnitude of this vector at t = 2 seconds.

Step-by-step explanation:

To find the speed of a particle at a given time t, you first need to determine the particle's velocity vector, which is the derivative of the position function r(t) with respect to time. The speed is the magnitude of this velocity vector. Given the position function r(t) = (1 - 2sin t, 1 - 2cos t, 0), we can calculate:

1. Derive the velocity function v(t) by differentiating each component of r(t) with respect to t.
2. Find the velocity at t = 2 seconds.
3. Calculate the speed as the magnitude of the velocity vector at that time.

The derivative of the position function is: v(t) = r'(t) = (-2cos t, 2sin t, 0).

Substituting t = 2 into the velocity function gives the velocity at that time. To find the speed, we take the square root of the sum of the squares of the velocity components. So the speed at time t = 2 seconds is: speed = √((-2cos(2))^2 + (2sin(2))^2 + 0^2).