A particle moves along the x axis in such a way that its...

A particle moves along the x-axis in such a way that its acceleration at time t for t ≥ 0 is given by a(t) = 4cos(2t). At time t = 0, the velocity of the particle is v(0) = 1 and its position is x(0) = 0. Write an equation for the position x(t) of the particle.

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Transcript

00:01 Here, we're given a situation where we know the acceleration and we know some initial conditions for the velocity and position, and we're asked to find the position in any given time.

00:10 So, for example, we start off with a of t, which is equal to four times the cosine of 2t.

00:18 If we want to find the velocity, we need to take the integral.

00:22 So v of t is going to be, we'll take the integral of cosine, which is sine of 2t, but we're going to have to divide by that.

00:33 2 because that comes from the chain rule.

00:36 So really this is 2 times the sign of 2t.

00:39 We could take the derivative and find that that is in fact the integral of 4 cosine of 2t.

00:44 But then we have to account for an initial condition.

00:48 So let's just say plus and we know that at time t equals 0, v is 1.

00:54 So v of 0 needs to be 1.

00:56 Well, if we don't add anything, v of 0 would be 0 because sine of 0 is 0.

01:01 So let's just add a 1 here.

01:03 So now we know v of 0 is 1.