{Use of Tech} A damped oscillator The displacement of an ...

Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
a. Find the time at which the object first passes the rest position, y = 0.

a. Find the time at which the object first passes the rest position, y = 0.

Accepted Answer

The height of a ball thrown vertically upwards from the ground is given by the equation H T equals 4 E to the negative T, cosine 4 T. Where H is the height in meters and T is the time in seconds. Find the first time at which the ball is at ground level. Now, we have 4 possible answers, being pi divided by 16, pi divided by 8, pi divided by 4, and pi divided by 12, all in seconds. So, we want, where this is at ground level, which means our height should be equal to 0. Let's set our equation equal to 00 equals 4 E to the negative T. Cosine 4 T. Now, 4 E to the negative T actually never equals 0, so we can actually divide this term out of our equation. We end up getting 0 equals cosine of 4 T. Now, cosine equals 0. At pi divided by 2. Plus in pie. So we can say 40. It equals to pi divided by 2, plus in pi. We divide both sides by 4. We end up getting T equals pi divided by 8 plus N multiplied. By pi divided by 4. No. We want the first place. Where T is equals to this value. So let's first use N equals 0. Where N is an integer We get T equals pi divided by 8. Plus 0 multiplied by pi divided by 4. This gives us T equals pi divided by 8. This is greater than 0. Which means the answer to our problem. It's pi divided by 8 seconds. Or answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

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