A particle, initially at rest, moves along the x-axis such that its acceleration at time t>0 is given by a(t)=cos(t). At the time t=0, its position is x=3.
How do I find the position function for the particle? I tried integrating the equation but got confused.
velocity is the integral of acceleration.
V= INT cos(t)= sinT + C
position is the integral of velocity..
position= INt (sinT+c)dt= -cosT+ CT+ D
So at t=0, position is zero
position=-cos0+c*O+ D so
3=-1+D and D=4
C cannot be determined without more information.
Thanks so much, I got that point but didn't that that was right. I guess I'll just leave as you explained. You've been super helpful. Thanks again!
What type of information would be needed? Does it matter that "the particle is moving along the x-axis where x(t) is the position of the particle at time t, x'(t) is its velocity, and x"(t) is its acceleration."?
Yes, it matters, but as you can see from the equations, unless you know the initial velocity, its position cannot be determined. If it comes shooting out of a gate at t=0, the initial position and acceleration can be the same, but its position will be a lot different if it starts from rest. So, you need either v(a) for some a, or p(a) for some a other than 0.
initially at rest means the initial velocity is zero.
To find the position function for the particle, you need to integrate the acceleration function with respect to time. Let's go through the process step by step:
1. Start with the given acceleration function: a(t) = cos(t).
2. To find the velocity function, integrate the acceleration function with respect to time. The integral of cos(t) with respect to t is sin(t). So, the velocity function v(t) is given by v(t) = sin(t) + C, where C is the constant of integration.
3. To find the position function, integrate the velocity function with respect to time. The integral of sin(t) with respect to t is -cos(t). So, the position function x(t) is given by x(t) = -cos(t) + D, where D is the constant of integration.
4. To determine the constant of integration, use the initial condition given in the problem. At t = 0, the position is x = 3. Substitute these values into the position function: 3 = -cos(0) + D. Since cos(0) = 1, we have 3 = -1 + D. Solving for D, we get D = 4.
5. Finally, substitute the value of D back into the position function. Therefore, the position function for the particle is x(t) = -cos(t) + 4.