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
v(t) = sin(t) + cos(t) + 3
v(t) = sin(t) + cos(t) + 2
v(t) = sin(t) - cos(t) + 3
v(t) = sin(t) - cos(t) + 4
v(t) = ∫ a(t) dt
= ∫ cost-sint dt
= sint+cost+C
Now plug in (0,3) to find C
To find the velocity, v(t), from the given acceleration, a(t), you need to integrate the acceleration function with respect to time.
Given that a(t) = cos(t) - sin(t) and v(0) = 3, we can integrate a(t) with respect to t to obtain v(t).
Step 1: Integrate a(t) to find v(t)
Integrating the acceleration function, we get:
v(t) = ∫[a(t)] dt
= ∫[cos(t) - sin(t)] dt
To integrate cos(t), you can use the formula for the integral of cosine:
∫ cos(t) dt = sin(t) + C1 <-- (where C1 is the constant of integration)
Similarly, to integrate sin(t), you can use the formula for the integral of sine:
∫ sin(t) dt = -cos(t) + C2 <-- (where C2 is another constant of integration)
So integrating both terms of a(t), we get:
v(t) = sin(t) + C1 - cos(t) + C2
= sin(t) - cos(t) + (C1 + C2)
Step 2: Determine the constants of integration
To determine the constants of integration (C1 and C2), we use the given initial condition v(0) = 3.
When t = 0, the expression for v(t) becomes:
v(0) = sin(0) - cos(0) + (C1 + C2)
= 0 - 1 + (C1 + C2)
= C1 + C2 - 1
Given that v(0) = 3, we have:
C1 + C2 - 1 = 3
Step 3: Find the specific expression for v(t)
We can rearrange the equation from Step 2 to solve for C1 + C2:
C1 + C2 = 3 + 1
C1 + C2 = 4
Substituting this back into the expression for v(t), we get:
v(t) = sin(t) - cos(t) + 4
Therefore, the correct answer is v(t) = sin(t) - cos(t) + 4.