A particle moves on a coordinate line with acceleration...

A particle moves on a coordinate line with acceleration $frac{d^2s}{dt^2} = 45sqrt{t} - frac{15}{sqrt{t}}$, subject to the conditions that $frac{ds}{dt} = 6$ and $s = 17$ when $t = 1$. Find the velocity $v = frac{ds}{dt}$ in terms of $t$ and the position $s$ in terms of $t$. The velocity $v = frac{ds}{dt}$ in terms of $t$ is $v = ?$. The position $s$ in terms of $t$ is $s = ?$.

Transcript

00:01 So for this question we are given an acceleration function, which i have written as a of t.

00:06 We are also told a condition for the velocity and position when time equals 1.

00:12 So our goal is to find both the velocity and position functions based on this information.

00:20 So first you have to find velocity, and this is done by integrating the acceleration.

00:30 Now it may be more beneficial to rewrite the acceleration.

00:36 Here, let's do that down here.

00:45 So i'm going to rewrite it as t to the 1 half for the square root minus 15, and this will be t to the negative 1 half.

01:03 Perfect, okay.

01:05 And now we can take the integral to get the velocity.

01:11 So you get 45, and now you'll get t to the 3 over 2, and then you divide by 3 over 2 as well.

01:24 When you divide by 3 halves, that's the same thing as multiplying by 2 thirds.

01:30 Okay, now what? minus 15, you will get t to the 1 half divided by 1 half.

01:40 You divide by 1 half, you multiply by 2.

01:43 And we're going to add a constant on the end because it's an indefinite integral.

01:50 The reason i'm calling it c1 is because we're going to have to integrate again when we evaluate position.

01:57 So i'm going to call that constant c2.

02:01 Okay, so we should simplify this a little bit.

02:05 45 times 2 thirds is 30.

02:09 So we have 30t to the 3 halves minus 30t to the 1 half plus c1.

02:23 So now with this simplified, in order to find c1, we need to use the initial condition.

02:30 In this case, if we plug in 1 for velocity, we should get 6.

02:35 So v of 1 equals 30 minus 30 plus c1...

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