1. 5.9 Quadratic Word Problems Working With Gravity

2. 5.9 Quadratic Word Problems Gravity is the force that causes objects to fall to the earth. Many famous scientists and mathematicians have studied gravity to try to form theories on how it works. One famous astronomer, Galileo went so far as to drop items off the leaning tower of Pisa to try to figure out how fast they went and if the size of the object mattered.

3. 5.9 Quadratic Word Problems What scientists eventually discovered is that all objects would fall at the same speed if they were in a vacuum* (a place that has no air pressure to slow an object down). It is friction from the air which causes objects to fall at different speeds, not the force of gravity. This is why a rock falls at one speed and a feather another speed. A feather has so much friction, that is falls a lot slower. Not this kind of vacuum! *A vacuum is an area that has no air pressure.

4. 5.9 Quadratic Word Problems Fact 1: An object falls at 32 feet per second (f/s) every second. If a rock falls off a cliff for 6 seconds, it is travelling at 192 f/s (6 x 32 = 192) If you jumped off a cliff into the water and you were in the air for 3 seconds, how fast were you travelling (velocity) when you hit the water? Answer: 96 f/s

5. 5.9 Quadratic Word Problems How to convert feet per second to miles per hour… This is not on the test, but is interesting to know. 96 feet Second 1 mile 5280 feet 60 seconds 1 minute 60 minutes 1 hour We are left with 96 x 1 x 60 x 60 miles 5280 hour After doing the math, the answer is 65.45 mph The conversion rate is.68

6. 5.9 Quadratic Word Problems Fact 2: If you throw an object straight up in the air (vertically), you can use the formula h = -16t 2 + vt + c to find Starting Height, Ending Height, Initial Velocity, and Time. Please become familiar with the variables… H: Ending Height T: Time V: Velocity (Speed) C: Starting Height Hint: The height of the ground is 0. If a rocket is launched from the ground, then C = 0; when it crashes, then H = 0.

7. 5.9 Quadratic Word Problems h = -16t 2 + vt + c Let’s see what we can learn from an equation... Mandy jumped off the diving board. The following equation represents Mandy’s dive: h = -16t 2 + 10t + 8 Here’s what we can tell just from the formula 1.How tall the diving board was. 2.How fast she jumped Starting height is “C”: 8 feet Velocity is “V”: 10 feet per second

8. 5.9 Quadratic Word Problems h = -16t 2 + vt + c You will have to work out a full gravity problem on your EOT. The next question is an example of the problem on the test. Jose shot an arrow into the air. It went straight up and came straight back down. It was in the air for 12 seconds. Please answer the following questions… 1. What was the initial velocity? 2. How high did the arrow get?

9. 5.9 Quadratic Word Problems h = -16t 2 + vt + c Jose shot an arrow into the air. It went straight up and came straight back down. It was in the air for 12 seconds. First Question: 1. What was the initial velocity? Use the gravity formula. First, get your variables… H = 0 (height when it hit the ground) T = 12 seconds V = I don’t know C = 0 (height when it started) h = -16t 2 + vt + c 0 = -16(12) 2 + v(12) + 0 0 = -2304 + 12v 2304 = 12v v = 192 f/s

10. 5.9 Quadratic Word Problems h = -16t 2 + vt + c Jose shot an arrow into the air. It went straight up and came straight back down. It was in the air for 12 seconds. We now know the velocity was 192 f/s. Now for the second question… 2. How high did the arrow get? H = I don’t know T = I’m not sure V = 192 f/s (Just figured it out) C = 0 (height when it started) Big problem! To find the maximum height, I need to know how long it took to get there (Time) Let’s look at our variables again…

11. 5.9 Quadratic Word Problems h = -16t 2 + vt + c Jose shot an arrow into the air. It went straight up and came straight back down. It was in the air for 12 seconds. We now know the velocity was 192 f/s. Now for the second question… 2. How high did the arrow get? H = I don’t know V = 192 f/s (Just figured it out) C = 0 (height when it started) Let’s look at our variables again… T = I’m not sure T = 6 Seconds h = -16t 2 + vt + c h = -16(6) 2 + 192(6) + 0 h = -576 + 1152 h = 576 feet

12. 5.9 Quadratic Word Problems h = -16t 2 + vt + c Jose shot an arrow into the air. It went straight up and came straight back down. It was in the air for 12 seconds. Let’s look get a picture of what is happening… If it took 12 second to go up and come down, how long did it take to go half way? (Half way is the maximum) My guess would be 6 seconds (half of 12 seconds) Out new T is 6 seconds.

13. 5.9 Quadratic Word Problems Please practice for this type of question by obtaining the 5.9 worksheet from your LF. If you want more practice, ask your LF for additional problems. Don’t take the EOT until you are ready.