Integrating to find velocity and position equations

In summary, the acceleration of a particle is described by the function a(t) = pt2-qt3, with p and q as constants. The velocity and position of the particle are initially zero. The velocity as a function of time is v(t) = p(t3/3) - q(t4/4), and the position as a function of time is x(t) = p(t4/12) - q(t5/20).
  • #1
uchicago2012
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0

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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  • #2
looks good to me
 

FAQ: Integrating to find velocity and position equations

How do you integrate to find velocity and position equations?

To integrate to find velocity and position equations, you will first need to have an understanding of calculus and its principles. Specifically, you will need to know how to perform integration using the fundamental theorem of calculus. Once you have this knowledge, you can use the equations for velocity and position and integrate them to find the desired equations.

What is the purpose of integrating to find velocity and position equations?

The purpose of integrating to find velocity and position equations is to determine the change in position and velocity over a certain period of time. This is useful in physics and engineering applications, such as calculating the trajectory of a projectile or the motion of a moving object.

Can you give an example of integrating to find velocity and position equations?

One example of integrating to find velocity and position equations is calculating the distance traveled by an object with a known acceleration and initial velocity. By integrating the acceleration equation, you can find the velocity equation, and by integrating the velocity equation, you can find the position equation.

Are there any limitations to integrating to find velocity and position equations?

One limitation to integrating to find velocity and position equations is that it assumes constant acceleration. In real-life situations, acceleration can change over time, making the equations less accurate. Additionally, this method may not work for complex systems with non-linear motion.

How do you know if you have integrated correctly to find velocity and position equations?

You can check if you have integrated correctly by differentiating the velocity and position equations that you have obtained. If the derivatives match the original equations for acceleration and velocity, then you have integrated correctly. Additionally, you can check your integration by using known initial conditions and comparing the results to the actual motion of the object.

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