Suppose a stone is thrown vertically upward from the edge of a...

Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t² + 32t + 48.
With what velocity does the stone strike the ground?

With what velocity does the stone strike the ground?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the velocity of the ball when it hits the ground. The ball is launched straight up from the top of a building with an initial speed of 20 m per second from a height of 30 m above the ground. The height in meters of the ball above the ground, 10 seconds after it is launched is given by H of T is equal to -1.5 T2 plus 20 T + 30, and we need to round our answer to the nearest whole number. We give 4 possible choices as our answers. For choice A, we have -54 m per second. For choice B, we have minus 24 m per second. For choice C, we have minus 64 m per second, and for choice D, we have minus 32 m per second. Now we need to find the velocity of the ball when it hits the ground. So, we first need to figure out the time at which the ball hits the ground, then we'll use that time to figure out the velocity. So, to figure out the time that the ball hits the ground, we're going to use our function HFT, which is the height of the ball above the ground T seconds after it is launched. So, since the ground is located at 0, we're gonna set HFT. Equal to 0, that means that 0 is going to be equal to minus 1.5 T 2 plus 20 T. Plus 30. So, I need to solve this equation for T, and so we have a quadratic equation here, so we can use our quadratic formula to find T. So, recall for your quadratic formula that we're gonna have T is equal to -20. Plus or minus the square root of the quantity of 20 squared. -4, multiplied by -1.5. Multiplied by 30. And that whole quantity is divided by the quantity of 2 multiplied by minus 1.5. So we just need to simplify this expression. So, the minus sign and the numerator and the denominator will cancel out. So this is going to be equal to 20. Plus or minus the square root. Of the quantity, we'll have 20 squared, which is 400. The two minus signs in the next terms will cancel out and give us a plus, so we'll have 400+, and when we multiply out the remaining terms, that's going to give us 180. And then that whole quantity is divided by 3. So simplifying further, this is going to be equal to 20 plus or minus the square root of 580. That quantity is divided by 3. So, when we use our calculator to evaluate the square root of 580, This is going to be equal to 20. Plus or minus? 24.08. And that quantity is divided by 3. So now we need to determine which sign that we need in order to get our time. So, we need our time to actually be a positive quantity. And we note that the 24.08 is actually larger than the value of 20. So, if we use the minus sign, we're going to get an overall negative time. So, we actually want to use the plus sign in order to get our time value. So we're going to have T is equal to 20 plus 24.08. That quantity is divided by 3. And when we evaluate this using our calculator, this is going to be approximately equal to 14.69 seconds. So now that we have the time at which the ball hits the ground, we now need to calculate our velocity as a function of time. So, we call that our velocity is given by the time derivative of our position equation. So that means that VFT. Which is our velocity as function of time, is going to be equal to the derivative with respect to T of our position, which is H of T. So this is going to be equal to the derivative with respect to T. Of the quantity of minus 1.5 T squared. Plus 20 tea. Plus 30. So taking a derivative of this expression, this is going to be equal to. For the first term, we're going to use our constant multiple and power rule to take that derivative. So we're going to have the constant, which is the minus 1.5, multiplied by the exponent, which is 2, and that gets multiplied by T raised to the quantity of 2 minus 1. So I need to subtract one from the X opponent. Then we will have plus. We'll do the same thing for the second term. We'll have our constant, which is 20, multiplied by the exponent, which in this case is 1. That gets multiplied by T raised to the quantity of 1 minus 1. Then for the last term, we're gonna be taking a derivative of a constant, which is 30, with respect to T. And so anytime we take a derivative of a constant recall that that is going to be 0, then we'll have plus 0 for the last term. So simplify this expression, this is going to be equal to -3 T plus 20. So now we just need to evaluate this when T is equal to 14.69. So we're gonna be looking for V of 14.69. And this is going to be equal to. -3, multiplied by the quantity of 14.69 in quantity plus 20. And evaluating this expression, this is going to be equal to. Minus 24.07. And we need to round this to the nearest whole number. So doing so, this is going to be approximately equal to -24. And this is going to have units of meters per second, since our height was given in meters and our time was given in seconds, and we took a derivative of our height with respect to time. So that's how we get the units of meters per second. So this is the answer that we are actually looking for, and if we compare that to our answer choices, the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Quadratic Functions

Velocity and Acceleration

Roots of a Function

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