An object with an initial position of x(0) = 3 has a velocity of v(t) =sin(t). Find its position at t =2.
x(t) = -cos t + c
now you know that x(0) = 3, so plug in t=0 to find c.
Then you can use t=2 to get x(2)
So the answer would be -cos(2)+4=4.416?
Looks good to me.
To find the object's position at a given time t, we can integrate its velocity function with respect to time.
The velocity function given is v(t) = sin(t).
To find the object's position, we need to integrate v(t).
∫ sin(t) dt = -cos(t) + C
Here, C represents the constant of integration.
To determine the constant of integration, we need to know the initial position of the object x(0) = 3.
When t = 0, x(t) = x(0).
Setting t = 0 in the equation above, we get:
-x(0) + C = 0
Since x(0) = 3, we have:
-3 + C = 0
Solving for C, we find that C = 3.
Therefore, the position function is:
x(t) = -cos(t) + 3.
To find the position at t = 2, substitute t = 2 into the position function:
x(2) = -cos(2) + 3.
Calculating the value, we find that the object's position at t = 2 is approximately 3.583.