Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = 2t + sin(t) and v(0) = 4
a(t) = 2t + sin(t)
v(t) = t^2 - cos(t) + c
given v(0) = 4
4 = 0 - cos 0 + c
4 = -1 + c
c = 5
v(t) = t^2 - cos t + 5
Well, well, well! Looks like we've got an object on the move! Now, to find the velocity, v(t), we need to integrate the acceleration, a(t). So let's get integrating and see where it takes us!
Integrating a(t) gives us v(t) = ∫(2t + sin(t)) dt
Now, bear with me, things might get a little wavy with that sin(t). But hey, that's what makes life interesting, right?
So, let's break it down! The integral of 2t with respect to t is t^2, and the integral of sin(t) is -cos(t). Put 'em together, and we've got:
v(t) = t^2 - cos(t) + C
But hold up, we're not done yet! We've got one last piece of the puzzle: the initial condition that v(0) = 4.
Plugging in t = 0 into our expression for v(t), we get:
4 = 0^2 - cos(0) + C
Simplifying that, we find that C = 5.
So the final answer is:
v(t) = t^2 - cos(t) + 5
And there you have it! The velocity of the object moving along the x-axis. Enjoy the ride!
To find the velocity, v(t), we can integrate the acceleration, a(t), with respect to time, starting from an initial velocity of v(0) = 4.
The acceleration, a(t), is given by a(t) = 2t + sin(t).
Integrating the acceleration, we get:
∫ a(t) dt = ∫ (2t + sin(t)) dt
Let's integrate each term separately:
∫ 2t dt = t^2 + C1
∫ sin(t) dt = -cos(t) + C2
Now, we can combine the two integrals:
∫ a(t) dt = t^2 - cos(t) + C
Since v(t) is the antiderivative of a(t), we have:
v(t) = t^2 - cos(t) + C
To find the constant C, we can use the initial velocity v(0) = 4:
v(0) = 0^2 - cos(0) + C
4 = 0 - 1 + C
C = 5
Therefore, the velocity function v(t) for the object moving along the x-axis with the given acceleration and initial velocity is:
v(t) = t^2 - cos(t) + 5
To find the velocity function, v(t), we need to integrate the acceleration function, a(t), with respect to time, t, and then apply the initial condition v(0) = 4.
Step 1: Integrate the acceleration function, a(t), with respect to time, t.
Integrating 2t + sin(t) with respect to t gives us:
∫ (2t + sin(t)) dt = t^2 - cos(t) + C
where C is the constant of integration.
Step 2: Apply the initial condition v(0) = 4.
Since v(0) denotes the velocity at t = 0, we can substitute t = 0 into the equation derived in Step 1:
v(0) = 0^2 - cos(0) + C
4 = 1 + C
C = 3
Step 3: Determine the velocity function, v(t).
Substituting C = 3 back into the equation derived in Step 1:
v(t) = t^2 - cos(t) + 3
Therefore, the velocity function, v(t), for an object moving along the x-axis with an acceleration function a(t) = 2t + sin(t) and initial condition v(0) = 4, is v(t) = t^2 - cos(t) + 3.