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Numerade Educator

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Problem 65 Hard Difficulty

A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, $ 450 $m above the ground.
(a) Find the distance of the stone above ground level at time $ t $.
(b) How long does it take the stone to reach the ground?
(c) With what velocity does it strike the ground?
(d) If the stonee is thrown downward with a speed of $ 5\;m/s $, how long does it take to reach the ground?

Answer

(a)$$
s(t)=-4.9 t^{2}+450
$$
(b)$$
9.58
$$
(c) Velocity $=-93.91 \mathrm{m} / \mathrm{s}$
(d) about 9.09 seconds

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Video Transcript

Hello. This problem mainly deals with the position function and the velocity function which happens to be the derivative of the position function. And our problem we are on top of a building that is 400 m tall, dropping a stone part A. We want to find the distance above ground at time. T since they did not give us a specific time. T. We are looking to write the formula in order to find the distance. An object is from the ground at any given time. So we keep what I have in black up at the top. Um notice for G we have both the 32 ft per second squared and a 9.8 m per second squared. However, our units are 450 m on the height of the building. So we're going to be using meters. So substitute in 9.8. We will not have the V. Not T. Because they did not give us an initial velocity. That means our hand is over hanging over the top of the building with our stone and we're just going to open our hand and let the stone go. We're not going to give it any type of velocity so we can skip the V. Not T. However s not is the initial height. And since we're on top of the building, Our height is 450 m. This gives us our formula for part A. Now for part B. We want to know how long it takes a stone to reach the ground. If it reaches the ground then its height above the ground is zero. So we're taking the formula from A. And actually sorry but I need to go back in here and if you multiply your negative one half times 9.8 you'll come up with um -4.9. Mhm -4.9 T square plus 4 50. And that's the formula I'll be using. So now we have zero equals negative 4.9 T squared Plus 4 50. We just will go about solving it. If you divide both sides by 4.9, you'll come up with T squared equals 91.8 3 7. When we take the square root of that we will have both the positive and negative value. However, our time in this situation cannot be negative. It has to be positive time after we drop the stone. So we're just going to keep the positive value which is Approximately 9.58 second to hit the ground to hit brown. Okay, move on to part C. We want to know what is the velocity when it hits the ground. The velocity function is a derivative of the position function. Since my position function is S. Of T equals negative 4.9 T square plus 4 50. I am going to take the derivative of that to get my velocity function And it will simply equal negative 9.80. This gives us a formula. It does not answer the problem. So we have to keep working now that we know that the formula for our velocity Is negative 9.80 and part B. We found out that it hits the ground at 9.58 seconds. So we're going to substitute that value 40 in this problem. I'll give us negative 9.8 Times 9.58. Which means the velocity when it hits the ground will be -93.88 meters per second or approximately negative 94 meters parasite. That takes us through part C. Moving on to part D. Now we're going to add on velocity instead of just opening our hand and allowing the stones to drop. We are going to throw it downward at a speed Of 5m/s so that's its initial velocity. So that will change the formula that we will have over here. So now our formula for our position Is negative 4.9 T sq minus five T. Yes. 4 50. Okay. Since we want to know how long it will take it to reach the ground. If we throw at that velocity, we're going to be substituting zero again for R. S. Of T. Then over here on the right, I'm going to use the quadratic formula to get straight to our values. Sovereign. This here. You already know the general formulas. I'm just going to straight substitute into it. If I continue working this out, I will have that two equals 5 plus or minus the square root of 88 45, 88 45 came from doing 25 minus the negative value. I'll get from multiplying four times negative 4.9 times for 50. And when I combine them I get 88 45 because there will be two negative. So that back part cancels out Over a negative 9.8. If you work both of them out, both the plus and the minus You'll get. The T. is equal to negative 10 107 seconds. Or T is equal to 9.087 Seconds which is approximately 9.1 2nd. Our time cannot be negative because if our time is negative, it happened before we drop the stone. So we're looking for the time after we drop the stone. So that's going to be the positive value. So we will not use the negative 10. Our solution for part D. See if I can make that a little bit better for you. Our solution for part D will be that it will hit The ground at 9.4.1 seconds or 9.087 seconds after you throw it