Integrating to find velocity and position equations

AI Thread Summary
The particle's acceleration is given by a(t) = pt² - qt³, with initial conditions of zero velocity and position. The velocity function is derived by integrating the acceleration, resulting in v(t) = p(t³/3) - q(t⁴/4) after determining the constant of integration is zero. The position function is obtained by integrating the velocity, yielding x(t) = p(t⁴/12) - q(t⁵/20), also with the constant set to zero. Both functions have been verified against the initial conditions, confirming their correctness. The approach to finding velocity and position through integration is clearly outlined.
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Homework Statement


The acceleration of a certain particle is a function of time: a(t) = pt2-qt3, where p and q are constants. Initially, the velocity and position of the particle are zero.
(a) What is the velocity as a function of time?
(b) What is the position as a function of time?

The Attempt at a Solution


I was just wondering if these looked correct:
a) I integrated a(t) to get
v(t) = p(t3/3) - q(t4/4) + c
then subbed in v(0) = 0 to find c = 0
so v(t) = p(t3/3) - q(t4/4)

b) Then I integrated v(t) to get
x(t) = p(t4/12) - q(t5/20) + c
then subbed in x(0) = 0 to find c =0
so x(t) = p(t4/12) - q(t5/20)
 
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