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.
v(t) = t2 + cos(t) + 3
v(t) = 2 + cos(t) + 1<- my answer
v(t) = t2 − cos(t) + 5
v(t) = t2 + sin(t) + 4
You have found da/dt. Instead, you are given a, so
v(t) = ∫v dt = t^2 - cos(t) + C
since v(0) = 4, C=5
v(t) = t^2 - cos(t) + 5
To find the velocity, v(t), of an object moving along the x-axis given the acceleration, a(t), and the initial velocity, v(0), we can use the integral of the acceleration function with respect to time.
In this case, the acceleration function is a(t) = 2t + sin(t), and the initial velocity is v(0) = 4.
To find the velocity function, we integrate the acceleration function:
∫(2t + sin(t)) dt = t^2 - cos(t) + C,
where C is the constant of integration.
Since we know that v(0) = 4, we can substitute t = 0 into the velocity function and solve for C:
v(0) = 0^2 - cos(0) + C = 4.
Simplifying this equation, we find:
1 + C = 4,
C = 3.
Therefore, the velocity function is:
v(t) = t^2 - cos(t) + 3.