Kinematics – the study of how things move When an object moves, what changes? position time Kinematics Constant Speed When you measure how fast an object’s position changes, you are measuring the speed. Speed – the rate at which an object changes its position 1

Suppose you are on a road trip and you have the cruise control set at 60 mph. In 1 hour, you would travel 60 miles In 2 hours, you would travel 120 miles In 10 hours, you would travel 600 miles The rate at which you are traveling is your constant speed. v = Δ x / t v = speed (m/s) Δ x = change in position (m) t = time (s) 2

1.A car travels at a constant speed of 20 m/s. a. How far will the car travel in 5 seconds? b. How far will the car travel in 7 seconds? c. How much time would it take for the car to travel 120 m? 2.A plane flies 400 miles in 30 minutes at a constant speed. What is the speed of the plane? 3. A turtle crawls 180 cm at a speed of 50 cm/hr. How long does it take the turtle to crawl that distance? 100 m 140 m 6 s 13.3 mi/min 3.6 hr 3

Average Speed Since a car’s speed changes frequently, it doesn’t have a constant speed. It has a variety of instantaneous speeds and it has an average speed. The instantaneous speed is the speed at one specific moment. The average speed is somewhere between the fastest instantaneous speed and the slowest instantaneous speed. The formula is the same as it is for constant speed. 4 The speedometer in your car measures the car’s speed at any one time. If you were driving the track below, why wouldn’t your speed be constant? v = Δ x / t

1.A runner speeds up between race markers x = 30.0 m and x = 80.0 m over a time of 12.0 seconds. What is the runner’s average speed between those markers? 2.The runner slows down between race markers x = 80.0 and x = m over a time of 15.0 seconds. What is the runner’s average speed between those markers? 3.What is the runner’s overall average speed? 4.17 m/s 2.0 m/s 2.96 m/s 5

Average Velocity There are times when it is important to know not just how fast something is traveling but also in which direction it is traveling. Velocity – the measurement of an object’s speed and direction Once again, the formula is the same v = Δ x / t An object moving forward has positive velocity. If that object returns in the opposite direction, it has negative velocity. Because velocity includes direction, it is a vector quantity. Speed is simply a scalar quantity 6

a.A cable car travels 80 m to the right in 7 seconds. What is the average velocity of the cable car? b. The same cable car then stops and travels 80 m to the left in 5 seconds. What is the average velocity of the cable car going the other way? a.A bungee jumpers falls 100 m toward the water in 4.5 seconds. What is his average velocity? b. When the cord recoils, the same bungee jumper takes 4.5 seconds to go back up 80 meters. What is his average velocity going up? m/s -16 m/s m/s m/s 7

There are very few objects in our everyday life that travel at a constant velocity. Most object experience a change in velocity as they move. When objects experience a consistent change in velocity (speed and direction), they experience an acceleration. There are three devices that can change a car’s velocity. What are they? Gas Pedal Brake Pedal Steering Wheel Acceleration 8

Acceleration – the rate at which an object’s velocity changes a = Δ v / t a = acceleration (m/s 2 ) Δ v = change in velocity (m/s) t = time (s) Suppose a car at rest accelerates at 20 m/s 2. Every second, the velocity of the car increases 20 m/s. After the 1 st second, the car would be going 20 m/s. After the 2 nd second, the car would be going 40 m/s. 9

1.A car accelerates along a straight road from rest to 75 m/s in 5.0 seconds. a. What is the car’s change in velocity? b. What is the car’s acceleration? 2.A rocket sitting on a launch pad blasts off with an acceleration of 150 m/s 2. a. How fast would the rocket be going after 3 seconds? b.How long would it take for the rocket to reach a speed of 750 m/s? c.How fast would the rocket be going 2 seconds after achieving a speed of 75 m/s? + 75 m/s 15 m/s m/s 5 s 375 m/s 10

Kinematic Equations When an object accelerates at a uniform rate, a set of equations can be used to analyze different characteristics about the object’s motion. Example: A sports car accelerates from a resting position at the starting line at a rate of 12 m/s 2 for 4.5 seconds. Suppose we want to know how fast the car is going after 4.5 seconds and how far it has traveled while accelerating… 11

slope = acceleration a = Δv / t a = v f – v i / t at = v f - v i “How fast?” – Equation 1 The line represents the change in velocity of the ball over time during impact (the acceleration). v f = v i + at v i = initial velocity v f = final velocity a = acceleration t = time Given what we know about the car, what would be the final velocity after accelerating 4.5 seconds? 54 m/s 12

“How far?” – Equation 2 area = displacement Δx = area A + area B Δx = ½ bh + bh Δx = ½(v f - v i )t + v i t Δx = ½v f t – ½v i t + v i t Δx = ½v f t + ½v i t The area under the line represents the change in position (displacement) of the golf ball. Δx = ½(v i + v f )t v i = initial velocity v f = final velocity ∆x = displacement t = time Given what we know about the car, what would be the displacement after accelerating 4.5 seconds? m 13

A sidewinder missile is dropped from a fighter jet traveling 200 m/s. If the missile immediately attains an acceleration of 15 m/s 2 when dropped, how fast is it going after 10 seconds? +350 m/s 14

A diver drops off of a cliff with an unknown height and falls toward the water. If he hits the water 3.40 seconds later traveling a velocity of 33 m/s, how high is the cliff? 56.1 m 15

A 3 rd Kinematic Equation: Substitute Equation 1 into Equation 2: Δx = ½(v i + v i + at)t Δx = ½(2v i + at)t Δx = (v i + ½at)t Δx = v i t + ½at 2 Equation 1 v f = v i + at Equation 2 Δx = ½(v i + v f )t 16

A 4 th Kinematic Equation: Substitute the rearranged Equation 1 into Equation 2: Δx = ½(v i + v f )t Δx = ½(v i + v f ) (v f – v i ) / a aΔx = ½(v i + v f ) (v f – v i ) 2aΔx = (v i + v f ) (v f – v i ) 2aΔx = v f 2 - v i 2 Equation 1 v f = v i + at v f - v i = at t = (v f – v i ) / a Equation 2 Δx = ½(v i + v f )t 17 v f 2 = v i 2 + 2aΔx

An airplane must reach a speed of at least 27.8 m/s in order to takeoff. It can accelerate at 2.00 m/s 2. If the runway is 150 m long, can this airplane reach the required velocity for takeoff? If not, what minimum length must the runway have? NO…194 m needed 18

Freefall Problems According to legend, Galileo Galilei climbed the Tower of Pisa in 1591 with two balls made of the same material but with different sizes (and different weights). He wanted to disprove Aristotle who believed that things fall at rates relative to their weights. To do that, he proceeded to drop both balls simultaneously and they struck the ground at the same time. This never actually happened, BUT Galileo believed that it would and he devoted much of his life to disproving Aristotle. He ultimately lost his job at the University of Pisa because of his rogue beliefs – but his beliefs would eventually be proven to be correct. Italian Mathematician and Physicist

Air resistance is negligible unless otherwise stated Objects that return to their starting position have a displacement of “0” Objects at their maximum height have a velocity of “0” Gravity is always acting on freefalling objects with an acceleration of 9.8 m/s 2 (g) When an object is falling in space without any forces acting on it except gravity, it is considered a “freefall” problem. To solve freefall problems, some things to remember include: 20

Suppose a ball is dropped from a tower 70.0 m high. How long would it take for the ball to hit the ground? 3.78 s 21

A stone is thrown from the edge of a cliff straight down with a speed of 3.00 m/s. If it hits the ground 4.00 seconds after it is thrown, how high is the cliff? 90.4 m 22

A person throws a ball upward into the air with an initial velocity of 15.0 m/s. Calculate how high it goes and how long the ball is in the air before it comes back to his hand. If the initial motion is upward, designate upward direction as positive and use – g for acceleration because it is directed downward m 3.06 s 23

Object speeding up Object slowing down Object moving in positive direction Acceleration is positive Acceleration is negative Object moving in negative direction Acceleration is negative Acceleration is positive Types of Acceleration 24

Projectile Motion A projectile is an object in freefall that is moving in two dimensions. Resultant trajectory is parabolic because: Horizontal motion is a constant velocity situation (no air resistance) Vertical motion is a changing velocity/constant acceleration situation (due to gravity) Because of the different situations, parabolic motion has to be broken down into x- and y- directions. 25

Horizontal and vertical motion act simultaneously but also independently. The horizontal component of the yellow ball’s motion does not affect the time it takes for it to move downward. Both balls would strike the floor simultaneously. 26

Typical Projectile Problem Setup “X” Variables for Constant Velocity in Horizontal Direction “Y” Variables for Constant Acceleration in Vertical Direction Δx = v c = t = Δ y = v i = v f = a = t = Horizontal Direction Equation Δx = v c t Vertical Direction Equations v f = v i + at Δy = v i t + ½at 2 v f 2 = v i 2 + 2aΔy Δy = ½(v i + v f )t 27

A movie stunt driver on a motorcycle speeds horizontally off a m high cliff. How fast must the motorcycle leave the cliff top to land on level ground below, 90.0 m from the base of the cliff? 28.2 m/s 28

A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff, and how far from its base did the diver hit the water? 44.1 m 5.4 m 29

30 Most projectiles do not start out moving in the horizontal direction. Most projectiles are shot, kicked, thrown, or launched at an angle. That velocity is made up of horizontal and vertical parts called components. If a football is kicked with a velocity of 25 m/s at an angle of 30 degrees, what is the horizontal and vertical components of that velocity? v c = v cos v i = v sin 30° 25 m/s 21.7 m/s 12.5 m/s Projectiles at an Angle

31 A stunt driver drives off of a 40 degree ramp at the top of a 25-m high cliff at a speed of 15 m/s. How long is he in the air and how far away from the base of the cliff does he land? 3.45 s 39.6 m

A football is kicked at an angle of 37.0 degrees with a velocity of 20.0 m/s. Calculate the maximum height and how far away it hits the ground m 39.2 m 32

If you know the horizontal and vertical components, you can also use trigonometry in the other direction to find the velocity at the angle (what you see). Move vectors so that they are tail- to-tip (vector addition) If they form a right angle, hypotenuse is the resultant velocity. 33 v 2 = v i 2 + v c 2 tan = v i / v c

A projectile is launched from ground level to the top of a cliff which is 195 m away and 155 m high. If the projectile lands on top of the cliff 7.6 s after it is fired, find the initial velocity of the projectile (magnitude and direction) m/s 66.2° above horizontal 34