Question

a catapult launches a boulder with an upward velocity of 92 m/s. the height of the boulder, h, in meters after t seconds is given by the function h=-5t^2+92t+16. how long does it take to reach maximum height? what is the boulders maximum height?

Answer 1

Do you know what curve that equation specifies?

Answer 2

No

Answer 3

It's a parabola, opening downward (because the coefficient of the \(t^2\) term is negative).
sorry for the crappy drawing!

Answer 4

The point of maximum height is the vertex of the parabola.

Answer 5

Do you know how to find the vertex?

Answer 6

No

Answer 7

Do you know what vertex form of the parabola is?

You can write a parabola that opens up or down (we will ignore parabolas that open right or left, and parabolas that are tilted) in two ways:

standard form:
\[y = ax^2+bx+c\]

vertex form:
\[y = a(x-h)^2+k\]

If you use the vertex form, the vertex of the parabola can just be read right out of the equation, and will be at \((h,k)\)

If you use the standard form, it's a little more involved to find the vertex. The x-coordinate will be at \[x=-\frac{b}{2a}\]and the y-coordinate will be whatever the value of \(y\) is when you plug in that value of \(x\).

Answer 8

our equation is in standard form: \(h = -5t^2+92t+16\)
which means that \(a = -5, b = 92, c = 16\) and our vertex will be at \[t = -\frac{b}{2a} = -\frac{92}{2(-5)} = \]

Answer 9

What is the value of \(t\)?

Answer 10

Idk?

Answer 11

It's a simple arithmetic problem:

\[t = -\frac{92}{2(-5)} = -\frac{92}{-10} = \frac{92}{10} = \]

Answer 12

What is 92 divided by 10?

Answer 13

9.2

Answer 14

Sorry my computer is kinda slow

Answer 15

Good, so at \(t=9.2 \text{ s}\) the rock reaches its maximum height.

Answer 16

Now, to find out what that maximum height actually is, you evaluate \[h(t) = -5t^2+92t+16\]when \(t = 9.2\) or\[h(9.2) = -5(9.2)^2+92(9.2)+16 = \]

Answer 17

Here's a graph of the entire flight of the boulder, showing height (y-axis) vs. time (x-axis):

Answer 18

A lot of these problems also ask you to find the time of flight (2*9.2 = 18.4 seconds)

Answer 19

So the maximum height would be 439.20?

Answer 20

You got it...

Answer 21

Thank you.

Answer 22

A bit of physics to go with the math:

the height formula has 3 components:

-5t^2 represents acceleration due to gravity (downward)
92t represents motion due to the initial upward velocity (upward)
16 represents the initial height when launched

at the apex of the trajectory, the second component will be cancelled out by the first, and after that the projectile will start falling because the acceleration due to gravity will have overwhelmed the upward velocity. right at the apex, the projectile is ever-so-briefly motionless as the curve changes from going up to going down. We can use calculus to find that spot as well. I won't walk you through it, but the first derivative of a curve, when equal to 0, is a point where the curve changes direction in this fashion. The first derivative of our height function is \[\frac{d}{dt}[h(t)] = 2*(-5)t^{2-1} + 1*92t^{1-1} = -10t+92\]If we set that equal to 0, we have\[0=-10t+92\]and solving for \(t\) we get \(10t=92\) or \(t=9.2\) just like we got by using the analytic geometry approach. The first derivative of the position (height) is the velocity (and the first derivative of the velocity is acceleration). At the top of the trajectory the velocity is instantaneously 0, so the physics and the math interpretations say the same thing.