Matching heights A stone is thrown with an initial velocity of...

Question

Matching heights A stone is thrown with an initial velocity of 32 ft/s from the edge of a bridge that is 48 ft above the ground. The height of this stone above the ground t seconds after it is thrown is f(t) = −16t²+32t+48 . If a second stone is thrown from the ground, then its height above the ground after t seconds is given by g(t) = −16t²+v0t, where v0 is the initial velocity of the second stone. Determine the value of v0 such that both stones reach the same high point.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says a soccer ball is kicked upwards from the top of a 60-foot tall building with an initial velocity of 40 ft per second. The height of the soccer ball above the ground 10 seconds after it is kicked is given by FFT, which is equal to -16 T2 + 40T + 60. A second soccer ball is kicked from the ground with an initial velocity of V knot. Its height above the ground 2 seconds after being kicked is GFT, which is equal to -16 T2 plus VT. Which should be the value of V knots so that both soccer balls reach the same maximum height. We're given 4 possible choices as our answers. For choice A, we have V knot is equal to 37.8 ft per second. For choice B, we have V knot is equal to 60 ft per second. For choice C, we have V knot is equal to 71.8 ft per second. And for choice D, we have V knot is equal to 73.8 ft per second. So this question asks us to find the initial velocity V knot, so that both soccer balls reach the same maximum height. Now, in order to find the knot, we first need to find the maximum height for our first soccer ball, using the information that we're given in the problem. Then we'll use that maximum height to determine the initial velocity V knot of the second soccer ball. So the first thing we need to do is determine the maximum height of our first soccer ball. So to do that, we'll need to determine the velocity as a function of time for that ball, and then set that velocity equal to 0 and solve for the time, and that will give us the time that it takes to reach the maximum height, since the maximum height occurs when our velocity is equal to 0. We'll then use that time to actually calculate the maximum height, which we'll then use later to find our initial velocity V0 for the second soccer ball. So the first thing we need to do is determine our velocity as a function of time for our first soccer ball. So we'll call that velocity VA. So basically, we'll call the first soccer ball ball A. So it's velocity VA as a function of time, is going to be equal to. And here we call that our velocity as a function of time is given by the time derivative of our position equation. So, in this case, we will have the derivative with respect to T. Of FFT, as FFT is our position as a function of time for our first soccer ball, ball A. So this is going to be equal to the derivative with respect to T of the quantity of minus 16 T squad, plus 40 T. Plus 60. So taking this derivative, this is going to be equal to, for the first term, we're going to get minus 32 T for its derivative, then we'll have plus 40 or the second term's derivative, plus 0 for the last term, since it's constant. And so simplifying this, this is going to be equal to minus 32 T. Plus 40. So now we need to set this equal to 0 and solve for T. So we'll have minus 32 T. Plus 40 is equal to 0. So, we'll add 32 T to both sides of the equation, so we'll have 40 is equal to 32 T. And dividing both sides by 32, we will get T is equal to 1.25 seconds. Though here our time is given in seconds. So now we'll have to use this time to figure out our maximum height. So, we're going to need to calculate F of 1.25. Which is equal to -16. Multiplying the quantity of 1.25 in quantity squared. What 40 multiplied by 1.25. Plus 60. And when we evaluate this expression, this is going to be equal to. 85 ft. And it's in feet since our height was given in feet. So now we can use this maximum height to determine our initial velocity V not for the 2nd soccer ball. So, we're going to have to use the second soccer ball's position equation, GFT, and we'll set GFT equal to 85, since we want the um Maximum height to be 85 for both balls. So we will have 85. is equal to -16T. Squared plus vnot. So here we actually need to determine the time at which our maximum height occurs for our 2nd soccer ball. So, we're gonna follow the same procedure, so we'll need to find its velocity as a function of time, set equal to 0, and that will be at the time at which our ball reaches the maximum height. So, we'll call the velocity for our 2nd ball, VB. The 2nd ball will be ball B, and this could be a function of time, and this is going to be equal to the derivative with respect to T of GFT. And this is going to be equal to the derivative with respect to T of the quantity of minus 16 T squad, plus V T. They're taking the derivative, this is going to be equal to minus 32, T plus B not. So now we need to set this expression equal to 0 and solve for T. So, we'll have minus 32 T. Plus V is equal to 0. So adding 32 T to both sides will have V is equal to 32 T. and then divide both sides by 32, we will get T is equal to V knot divided by 32. So we'll substitute this value back into our expression for GFT, and so we'll have 85. is equal to. -16, multiplied by the quantity of Vno divided by 32 in quantity squared. Plus, V knot, multiplied by the quantity of V knot divided by 32. Now, we can combine the two terms on the right-hand side with a common denominator of 64, and so we will have 85. is equal to V squared divided by 64. And we now need to solve 4 V knots. So we'll multiply both sides by 64, and so we will get 5440. is equal to V squared. So now we just need to take the square root to both sides, and we'll get V knot is equal to the square root of 5440, which is approximately equal to. 73.8. This has units of feet per second, since our height was given in feet and our time was given in seconds. So this is the answer to the problem that we were looking for. And so now, if we take our final answer here and compare it to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Quadratic Functions

Vertex of a Parabola

Initial Velocity and Its Impact

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