Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = cos(t) − sin(t) and v(0) = 3.
if a(t) = cos(t) − sin(t)
v(t) = sin(t) + cos(t) + c
given: v(0) = 0
3 = sin0 + cos0 + c
3 = 0 + 1 + c
c = 2
v(t) = sin(t) + cos(t) + 2
To find the velocity, v(t), we can integrate the given acceleration function, a(t).
Given:
a(t) = cos(t) - sin(t)
To integrate a(t), we can integrate each term separately:
∫ a(t) dt = ∫ (cos(t) - sin(t)) dt
The integral of cos(t) is sin(t), and the integral of sin(t) is -cos(t).
So, integrating each term:
∫ a(t) dt = ∫ (cos(t) - sin(t)) dt
= ∫ cos(t) dt - ∫ sin(t) dt
= sin(t) - (-cos(t))
= sin(t) + cos(t)
Now, the integral of a(t) is v(t). Hence,
v(t) = sin(t) + cos(t)
Since we have an initial velocity, v(0) = 3, we can substitute t = 0 into the derived function to find the constant of integration.
v(0) = sin(0) + cos(0) = 0 + 1 = 1
Since the given initial velocity is 3, we need to add the constant of integration, C, to the derived function:
v(t) = sin(t) + cos(t) + C
To find the value of C, we can substitute t = 0 and v(0) = 3 into the derived function:
3 = sin(0) + cos(0) + C
3 = 1 + 1 + C
C = 3 - 2
C = 1
Therefore, the velocity function, v(t), for the given object moving along the x-axis is:
v(t) = sin(t) + cos(t) + 1