A particle moves along the x-axis so that its position as a function of time is given by the equation, find the velocity (as a function of time) and the acceleration (as a function of time). (t for time)
x = 10t^5 - 6t^3 + 13t + 15
v(t) = dx/dt = 50t^4 - 18t^2 + 13
a(t) = dv/dt = 200t^3 - 36t
To find the velocity of the particle as a function of time, we take the derivative of the position equation with respect to time.
Given the position equation:
x = 10t^5 - 6t^3 + 13t + 15
We can find the velocity by differentiating x with respect to t:
v = dx/dt
Differentiating each term of the equation separately:
d/dt (10t^5) = 50t^4 (using the power rule, where power n becomes n * t^(n-1))
d/dt (-6t^3) = -18t^2 (using the power rule)
d/dt (13t) = 13 (the derivative of a linear term is simply its coefficient)
d/dt (15) = 0 (a constant term has a derivative of 0)
Combining the results:
v = 50t^4 - 18t^2 + 13
Therefore, the velocity as a function of time is given by v = 50t^4 - 18t^2 + 13.
To find the acceleration of the particle as a function of time, we differentiate the velocity equation with respect to time.
a = dv/dt
Differentiating each term of the velocity equation:
d/dt (50t^4) = 200t^3 (using the power rule)
d/dt (-18t^2) = -36t (using the power rule)
d/dt (13) = 0 (a constant term has a derivative of 0)
Combining the results:
a = 200t^3 - 36t
Therefore, the acceleration as a function of time is given by a = 200t^3 - 36t.