The position of a particle moving along the x-axis is given by x = 3.49t2 – 2.21t3, where x is in meters and t is in seconds. What is the position of the particle when it achieves its maximum speed in the positive x-direction?
To find the position of the particle when it achieves its maximum speed in the positive x-direction, we need to determine the velocity function and then find the time when the velocity is at its maximum.
First, we need to find the velocity function by taking the derivative of the position function with respect to time (t). So, let's calculate it step by step:
1. Differentiate the position function:
dx/dt = d/dt (3.49t² - 2.21t³)
2. Apply the power rule of differentiation:
dx/dt = 2 * 3.49t - 3 * 2.21t²
3. Simplify the equation:
dx/dt = 6.98t - 6.63t²
Now that we have the velocity function (dx/dt), we can find the time when the velocity is at its maximum.
To find the maximum of a quadratic function, we know that the maximum point occurs at the vertex of the parabola. The vertex is given by the formula:
t = -b / (2a)
In our case, a = -6.63 and b = 6.98.
t = -6.98 / (2 * -6.63)
Simplifying the equation further:
t = 0.526 seconds
Now that we have the time (t = 0.526 seconds) at which the velocity is at its maximum, we can substitute this value back into the position equation to find the position of the particle.
x = 3.49t² - 2.21t³
Substituting t = 0.526:
x = 3.49(0.526)² - 2.21(0.526)³
Calculating further:
x ≈ 1.416 meters
Therefore, the position of the particle when it achieves its maximum speed in the positive x-direction is approximately 1.416 meters.