Throwing a stone Suppose a stone is thrown vertically upward f...

Question

Throwing a stone 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 $s (t) = - 16 t^{2} + 32 t + 48$ .
d. When does the stone strike the ground?

d. When does the stone strike the ground?

Accepted Answer

Welcome back, everyone. In this problem, from the edge of a cliff 48 meters above the ground, a ball is vertically thrown upward at a velocity of 16 meters per second. The function h of t equals negative 4 t squared plus 16 t plus 48 represents its height above the ground after it is thrown, where t is time in seconds. We want to determine the time at which the ball hits the ground. For our answer choices, a is 6 seconds, b is 2 seconds, c is 8 seconds, and the d is 4 seconds. Now, let's try to sketch this problem to make sense of it. So we're throwing a ball from the edge of a cliff. Okay? And from our cliff, we know that our cliff is 48 meters above the ground. Now, since we're throwing the ball up and it lands on the ground, then it would probably look something like this because it would form a parabola. And we have a function that models a parabola which would be the function of the height of the ball above the ground where t is time in seconds. So we have h of t equals negative 4 t squared plus 16 t plus 48. What else do we know? Well, we know it's being thrown upward at a velocity of 16 meters per second and we want to figure out the time at which the ball hits the ground. So, let's think about that. On our diagram, where would that be? Well, that would be down here. Okay? And, that would be our ball hitting the ground. Now, let's think. When our ball hits the ground, what do we know about the parameters? We don't know the time. The time is what we don't know but we know on the ground, its height would be equal to 0. So when the ball hits the ground, its height, h of t equals 0 and h of t equals 0 because h of t represents its height above the ground. So since we have a value for h of t, then we should be able to solve for a T in our function. So let's go ahead and do that. So. At H of T. Being equal to 0, so the ball hits the ground, then substituting it in our function, 0 equals negative 4 t squared plus 16 t plus 48. Now, notice that all of these terms have a common factor of 4. So we can factor 4 out of each term in our equation. So dividing through let me make a note of that. So dividing, let me write it a little bit further below dividing through our equation by negative 4. Let's use the coefficient of the squared term here Then negative well, 0 divided by negative 4 equals 0. Negative 4 t squared divided by negative 4, we'll have 16 t divided by negative 4 and 48 divided by negative 4. Negative 4 divided by negative 4 leaves us with t squared. 16 t divided by negative 4 gives us negative 4 t, And 48 divided by negative 4 equals negative 12. Now we can rewrite this with our expression t squared minus 4 t minus 12 equals 0 on the left hand side. And now we should be able to solve for t. This is a quadratic equation and we can first check if we can factorize to solve for the value of t. In other words, can we write t squared minus 4 t minus 12 where we have an expression that has a common fact where both expressions of t rather have common factors of 12? Well, we can rewrite this as t squared minus 6 t plus 2 t minus 12 equals 0. And now factoring each expression in our first two expressions, t is a common factor. So t into t squared goes t times, t into negative 6 t goes negative 6. Likewise, common factor between 2 t and negative 12 is 2. 2 into 2 t goes t times, 2 into negative 12 was negative 6 times. Now both terms have a common factor of t minus 6. So we can combine the outside terms, which means t plus 2 multiplied by t minus 6 equals 0, which means that either t plus 2 equals 0 or t minus 6 equals 0 because one of them has to be 0 for their product to be 0. Either one of them or both of them. Now if t plus 2 equals 0, that means t equals negative 2. And if t minus 6 equals 0, that means t equals 6. Now, since the problem requires time, when we think about it, time can't be negative. We can only have a positive value of t. Therefore, t could not be equal to negative 2. T must be equal to 6 seconds. When we look back on our answer choices, that would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Quadratic Functions

Finding Roots

Projectile Motion

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