1. Notes on Motion VI Free Fall A Special type of uniform acceleration

2. Free Fall When an object is only allowed to move under the influence of gravity. When you drop a ball, it is in free fall! When you throw a ball downward, it is in free fall! (After it is released) When you throw a ball upward, it is in free fall! (After it is released)

3. Free Fall When an object is in free fall, it has a uniform acceleration. Objects in free fall are ALWAYS accelerated downward by gravity. On Earth, objects in free fall are accelerated downward at a rate of 9.8 m/s/s (or 9.8 m/s 2 ). When we are working with formulas we use “g” to represent the acceleration due to gravity. g = 9.8 m/s 2 In the English System of measurement g = 32 ft/s 2.

4. Free Fall Speed/Velocity If an object, like a ball, is dropped, then we say it is in free fall from rest. Since down is generally considered to be the negative direction, the acceleration of the ball is: a = -g = -9.8 m/s 2 a This means that the ball’s velocity is changing by -9.8 m/s each second. Since the ball is also moving in the negative direction, the same direction as its acceleration, the ball is speeding up at a rate of 9.8 m/s each second. v Remember, at rest means the initial velocity is: 0-m/s

5. Free Fall Speed/Velocity Since we know that the acceleration of a ball that is dropped is -9.8 m/s 2, we can find its speed and velocity at any time after it is dropped. t = 0-s, speed = 0-m/s & v = 0-m/s t = 4-s speed = 39.2-m/s v = -39.2-m/s t = 1-s speed = 9.8-m/s & v = -9.8-m/s t = 2-s speed = 19.6-m/s v = -19.6-m/s t = 3-s speed = 29.4-m/s v = --29.4-m/s Initially the speed and velocity are both 0-m/s After 1-s, its velocity changes by -9.8-m/s: 0+ -9.8-m/s = -9.8-m/s Its speed is just 9.8-m/s. After 2-s, its velocity changes by another -9.8-m/s: -9.8-m/s + -9.8-m/s = -19.6-m/s Its speed is just 19.6-m/s After 3-s, its velocity changes by another -9.8-m/s: -19.6-m/s + -9.8-m/s = -29.4-m/s Its speed is just 29.4-m/s. After 4-s, its velocity changes by another -9.8-m/s: -29.4-m/s + -9.8-m/s = -39.2-m/s Its speed is just 39.2-m/s. It turns out that your speed was just the size of the velocity (no negative). To get the velocity you multiplied -9.8 by time: v = -9.8t

6. Free Fall Speed/Velocity How does this change if instead of dropping the ball, I throw the ball downward at 5-m/s t = 0-s, speed = 5-m/s & v = -5-m/s t = 4-s speed = 44.2-m/s v = -44.2-m/s t = 1-s speed = 14.8-m/s & v = -14.8-m/s t = 2-s speed = 24.6-m/s v = -24.6-m/s t = 3-s speed = 34.4-m/s v = --34.4-m/s Initially the speed of the ball is 5-m/s but the velocity of the ball is -5-m/s After 1-s, velocity changes by -9.8-m/s: -5-m/s + -9.8-m/s = -14.8-m/s Its speed is just 14.8-m/s. After 2-s, its velocity changes by another -9.8-m/s: -14.8-m/s + -19.6-m/s = -24.6-m/s Its speed is just 24.6-m/s. After 3-s, its velocity changes by another -9.8-m/s: -24.6-m/s + -9.8-m/s = -34.4-m/s Its speed is just 34.4-m/s. After 4-s, its velocity changes by another -9.8-m/: -34.4-m/s + -9.8-m/s = -44.2-m/s Its speed is just 44.2-m/s. The speed of the ball is just the size of the velocity (no negative) To get the velocity you multiplied -9.8 by time and added the initial velocity: v = -5 + -9.8t The ball still accelerates at -9.8-m/s 2, but you need to add the original 5-m/s in to your calculations.

7. Free Fall Speed/Velocity If an object, like a ball, is thrown upward, then we still say it is in free fall. Since down is generally considered to be the negative direction, the acceleration of the ball is still: a = -g = -9.8 m/s 2 a This means that the ball’s velocity is changing by -9.8 m/s each second for the entire time. But now the ball is initially moving in the positive direction, the opposite direction of its acceleration. This means the ball will slow down at a rate of 9.8 m/s each second, at least until it stops. v At this point the ball will act just like the dropped ball from before and speed up at 9.8 m/s each second.

8. Free Fall Speed/Velocity Lets say I throw this ball upward at 39.2 m/s. Initially the speed of the ball is 39.2-m/s and the velocity of the ball is 39.2-m/s After 1-s, its velocity changes by -9.8-m/s: 39.2-m/s + -9.8-m/s = 29.4-m/s Its speed is also 29.4-m/s. The ball still accelerates at -9.8-m/s 2, but you need to add the original 39.2-m/s in to your calculations. Time (s)Speed (m/s) Velocity (m/s) 0 39.2 2 3 4 5 6 7 1 8 29.4 19.6 9.8 0 19.6 29.4 39.2 29.4 19.6 After 2-s, its velocity changes by -9.8-m/s: 29.4-m/s + -9.8-m/s = 19.6-m/s Its speed is also 19.6-m/s. After 3-s, its velocity changes by -9.8-m/s: 19.6-m/s + -9.8-m/s = 9.8-m/s Its speed is also 9.8-m/s. 9.8 After 4-s, velocity changes by -9.8-m/s: 9.8-m/s + -9.8-m/s = 0-m/s Its speed is also 0-m/s. THE BALL HAS STOPPED!!! 0 After 5-s, velocity changes by -9.8-m/s: 0-m/s + -9.8-m/s = -9.8-m/s THE BALL HAS CHANGED DIRECTION!!! Its speed is just 9.8-m/s. -9.8 After 6-s, its velocity changes by -9.8-m/s: -9.8-m/s + -9.8-m/s = -19.6-m/s Its speed is just 19.6-m/s. -19.6 -29.4 -39.2 After 7-s, velocity changes by -9.8-m/s: -19.6-m/s + -9.8-m/s = -29.4-m/s Its speed is just 29.4-m/s. After 8-s, velocity changes by -9.8-m/s: -29.4-m/s + -9.8-m/s = -39.2-m/s THE OPPOSITE OF ITS ORIGINAL VELOCITY Its speed is just 39.2-m/s. ITS ORIGINAL SPEED At the ball’s maximum height the speed and velocity are both 0-m/s It takes the same amount of time for the ball to reach its maximum height as it does to get back to its original height (4-s). When the ball gets back to its original height, it has the same speed but the opposite direction as it did when it was thrown.

9. Free Fall Speed/Velocity In general, the formula we use to find the velocity of an object in free fall is: But since we know the acceleration is the acceleration due to gravity, we can change this to: Rewriting to find time, you get: The speed of an object in free fall is just the velocity without the sign.

10. Example Problem 1 A cliff diver dives from the top of a cliff. If he is able to fall for 10-s, what will his speed be when he hits the water? Useful Formulas Know Find Formula Solution v i = 0-m/s t = 10-s Free Fall v f = ? Since we are only asked for speed, we can ignore the (-), so the speed is 98-m/s.

11. Example Problem 2 A ball is thrown upward into the air at 35-m/s. How long will it take the ball to reach its maximum height? Useful Formulas Know Find Formula Solution v f = 0-m/s v i = 35-s Free Fall t = ? What would the hang time of this ball be? Since it takes 3.6-s to reach its maximum height, it will take an additional 3.6-s to reach the ground again. 3.6 + 3.6 = 7.2-s

12. Distance in Free Fall When an object that started from rest had a uniform acceleration, we had an equation for the distance it travelled in a given time: But since we know the acceleration of the object is “g” (9.8-m/s 2 ), we can change this to: If we need to find the time it takes an object to fall a certain distance, we can solve this equation for t:

13. Example Problem 3 A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How far will this rock have fallen in 5-s? Useful Formulas Know Find Formula Solution t = 5-s Free Fall d = ? How far from the ground will the rock be at this time? h = 381 - 122.5 = 258.5-m

14. Example Problem 4 A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How long will it take for the rock to hit the ground? Useful Formulas Know Find Formula Solution d = 381-m Free Fall t = ?

15. Example Problem 5 A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How fast will the rock be travelling just as it hits the ground? Useful Formulas Know Find Formula Solution t = 8.8-s From Example 4 Free Fall v f = ? Since we are only asked for speed, we can ignore the (-), so the speed is 86.2-m/s. FYI: 86.2-m/s is approximately 194-mph. That could do some damage.

16. Finding Maximum Height When an object is thrown upward, we know that the time it takes for it to reach its maximum height is the same as the time for it to fall back to the ground from that height. We can use this information to help us find the maximum height of the object. First Find the time it takes to reach its maximum height using the formula: Second Use the time found in the first step to calculate the distance it falls from its maximum height. Use: We also know that its velocity at its maximum height is 0.

17. Example Problem 6 A rifle fires a bullet at 896-m/s. If the bullet is fired straight up, find its maximum height. Useful Formulas Know Find Formula Solution v i = 896-m/s vf = 0-m/s Free Fall t = ? First find the time to maximum height: NOW USE THIS TIME TO FIND THE MAXIMUM HEIGHT

18. Example Problem 6 Continued A rifle fires a bullet at 896-m/s. If the bullet is fired straight up, find its maximum height. Useful Formulas Know Find Formula Solution t = 91-s Free Fall d = ?