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?
take the derivative of position to get speed. Then take the derivative of speed, set to zero, and find t for the max speed. Then put that t into the position equation.
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To find the position of the particle when it achieves its maximum speed in the positive x-direction, we need to find the first derivative of the position function and determine when it equals zero. When the derivative is zero, it indicates a point where the slope of the position function changes, which corresponds to a maximum or minimum.
Let's differentiate the position function with respect to time (t):
dx/dt = 6.98t - 6.63t^2
To find when the particle achieves its maximum speed, we need to set the first derivative equal to zero and solve for t:
0 = 6.98t - 6.63t^2
Now, rearrange the equation to get a quadratic equation:
6.63t^2 - 6.98t = 0
Factor out t:
t(6.63t - 6.98) = 0
Now, set each factor equal to zero and solve for t:
t = 0 or 6.63t - 6.98 = 0
From t = 0, we can disregard this value because it represents the initial time when the particle starts moving.
Solve the second equation for t:
6.63t - 6.98 = 0
Add 6.98 to both sides:
6.63t = 6.98
Divide both sides by 6.63:
t = 6.98 / 6.63
t ≈ 1.053 seconds
Now that we have the time at which the particle achieves its maximum speed, we can substitute this value back into the position function to find the position:
x = 3.49t^2 – 2.21t^3
x = 3.49(1.053)^2 – 2.21(1.053)^3
Calculate this expression:
x ≈ 3.49(1.110) – 2.21(1.170)
x ≈ 3.8809 – 2.5827
x ≈ 1.2982
Therefore, the position of the particle when it achieves its maximum speed in the positive x-direction is approximately 1.2982 meters.