Linear Motion or One Dimensional Kinematics SP 1

Linear Motion or One Dimensional Kinematics SP 1. Students will analyze the relationships between force, mass, gravity, and the motion of objects. a. Calculate average velocity, instantaneous velocity, and acceleration in a given frame of reference. b. Compare and contrast scalar and vector quantities. c. Compare graphically and algebraically the relationships among position, velocity, acceleration, and time.

Introduction • Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. • Kinematics is a branch of mechanics. The goal of any study of kinematics is to develop sophisticated mental models which serve to describe (and ultimately, explain) the motion of real-world objects.

Introduction • Physics is a mathematical science. The underlying concepts and principles have a mathematical basis. • Throughout the course of our study of physics, we will encounter a variety of concepts which have a mathematical basis associated with them. • It is imperative that you have a functional scientific calculator with you from now on – NOT A CELL PHONE CALCULATOR, as those will NOT be permitted on tests/quizzes.

Scalars and Vectors • Scalars are quantities which are fully described by a magnitude alone. – Magnitude refers to the size of a measurement • Vectors are quantities which are fully described by both a magnitude and a direction. – Direction can be given as a movement through space (N S E W) or within a coordinate plane (+ or -). – Vectors are represented visually with arrows pointing in the direction of movement.

Scalars and Vectors Check Your Understanding: Categorize each quantity as being either a vector or a scalar. QUANTITY CATEGORY a. -5 cm b. 256 bytes c. 30 m/sec, East d. 80 degrees Celsius e. 4000 Calories f. 7 km, 50° NW

MATH ALERT! • Because vectors involve direction, the problems will often involve combining values that either are or are not in the same direction. • Vectors are always connected head to tail, and it does not matter in which order.

MATH ALERT! • For single axis (all X or all Y) measurements, we connect the vectors head to tail and simply add the values together, retaining the sign to represent direction.

MATH ALERT! • For dual axis (both X and Y) measurements, we connect the component vectors head to tail to create a right triangle. • Connect the initial to the final position with a diagonal vector (i. e. the hypotenuse), called the resultant. • Use the Pythagorean theorem to calculate the magnitude of the resultant vector. • Use trig functions to calculate the direction of the resultant vector.

MATH ALERT!

Distance and Displacement • Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion. – It can never be negative, never decreases, and can have multiple directions. • Displacement is a vector quantity which refers to "how far out of original position an object is“ – It is the object's overall change (Δ) in position Δ = – Displacement can be negative, represents the final – shortest path between two points, and has a initial singular direction.

Distance and Displacement

Distance vs. Displacement Check Your Understanding: 1. You walk 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. A. What is the distance covered? _____ m B. What is the displacement? _____ m _____

Distance vs. Displacement Check Your Understanding: 2. You walk 3 meters East, 7 meters South, 7 meters West, and finally 7 meters North. A. What is the distance covered? _____ m B. What is the displacement? _____ m _____

Distance vs. Displacement Check Your Understanding: 3. You walk 3 meters East, 4 meters South, 5 meters West, and finally 6 meters North. A. What is the distance covered? _____ m B. What is the displacement? _____ m _____

Speed and Velocity • Speed is a scalar quantity which refers to "how fast an object is moving. “ – Speed can be thought of as the rate at which an object covers distance, or distance per time. – A fast-moving object has a high speed; a slowmoving object has a low speed. – An object with no movement at all has a zero speed.

Instantaneous Speed and Average Speed • Instantaneous speed is the speed at any given instant in time. – What the speedometer reads when you look at it • Average speed is the average of all instantaneous speeds – The average of an infinite number of speedometer readings during a trip – Found simply by a total distance/total time ratio

Speed and Velocity • Velocity is a vector quantity which refers to "the rate at which an object changes its position. “ – Velocity can be thought of as the rate at which an object covers displacement, or displacement per time. • Average velocity is calculated by a total displacement/total time ratio, with consideration of direction.

Average Speed vs. Average Velocity Check Your Understanding: 1. While on vacation, you traveled from Augusta to Atlanta for a total distance of 234 km. Your trip took 2. 5 hours. What was your average speed? ____ km/hr Can you determine average velocity?

Average Speed vs. Average Velocity Check Your Understanding: 2. You walk 3 meters East, 7 meters South, 7 meters West, and finally 7 meters North. The entire motion lasted for 24 seconds. Average speed = _______ m/s Average velocity = _____ m/s _____

Average Speed vs. Average Velocity Check Your Understanding: 3. Joe is driving to Macon. The first leg of the trip is via I-20 W, where he travels 144 km to Madison. He then takes US 129 S for 70 km to Macon. The trip takes 2. 25 hr. Average speed = _______ m/s Average velocity = _____ m/s _____

More Problems 1. A runner makes one lap around a 200 m track in a time of 25. 0 s. What was the runner's average speed? 2. If a car moves with an average speed of 60 km/hr, how far will it travel if it continues this average rate for 4 hrs? 3. A bullet is shot from a rifle with a speed of 720 m/s. What time is required for the bullet to strike a target 3240 m away?

More Problems 4. A bird is heading toward a feeder for 11 m north. It then travels east for 8 m and lands on a perch. The whole trip takes 25 s. What is the bird’s average velocity?

Position – Time Graphs • The slope of the line on a position versus time graph is equal to the velocity of the object. xf – xi tf - t i • Check Your Understanding: Determine the velocity (i. e. , slope) of the object as portrayed by the graph.

Position – Time Graphs

Position – Time Graphs Constant Velocity Slow, Positive Constant Velocity Fast, Positive Constant Velocity Slow, Negative Constant Velocity Fast, Negative

Acceleration • Acceleration is a vector quantity which is defined as the rate at which an object changes its velocity. – An object is accelerating if it is changing its velocity. • For objects with a constant acceleration, the distance of travel is directly proportional to the square of the time of travel.

Acceleration

Acceleration • The direction of the acceleration vector depends on two things: Øwhether the object is speeding up or slowing down Øwhether the object is moving in the + or - direction • The general RULE OF THUMB is: ØIf an object is slowing down, then its acceleration is in the opposite direction of its motion.

Animation

Average Acceleration Check Your Understanding: 1. Use the equation for acceleration to determine the acceleration for the following two motions. • Acceleration A = _____ m/s/s or m/s 2 • Acceleration B = _____ m/s/s or m/s 2

Average Acceleration Check Your Understanding: 2. A roller coaster car rapidly picks up speed as it rolls down a slope. As it starts down the slope, its speed is 4 m/s. Three seconds later, at the bottom of the slope, its speed is 22 m/s. What is its average acceleration?

Average Acceleration Check Your Understanding: 3. You are traveling in a car that is moving at a velocity of 20 m/s. Suddenly, a car 10 meters in front of you slams on its brakes. At that moment, you also slam on your brakes and slow to 5 m/s. Calculate the acceleration if it took 2 seconds to slow your car down.

More Problems 4. Drag racers are able to accelerate at 12. 5 m/s 2 from rest to 100 m/s before crossing the finish line. How much time elapses during the run? 5. A bus decelerates at -8. 3 m/s for 3. 0 s to pick up passengers. What is the bus’s velocity change? 6. You ride for 2 hours on cruise control at 70 mi/hr. What is your average acceleration?

Kinematic Equations v = Δx t a = Δv t vf = vi + at vf 2 = vi 2 + 2 aΔx Δx = vit + ½at 2 Δx – displacement (how far? ) t – time (how long? ) a – acceleration vi – initial velocity vf – final velocity

Kinematic Example Ima Hurryin is approaching a stoplight moving with a velocity of +30. 0 m/s. The light turns yellow, and Ima applies the brakes and skids to a stop. If Ima's acceleration is -8. 00 m/s 2, then determine the displacement of the car during the skidding process.

Free Fall and Gravitational Acceleration • A free-falling object is one which is falling under the sole influence of gravity. This definition of free fall leads to two important characteristics about a freefalling object: ØFree-falling objects do not encounter air resistance ØAll free-falling objects (on Earth) accelerate downwards at a rate of -9. 8 m/s 2 ØThis rate is commonly referred to as g

Free Fall and Gravitational Acceleration • All of the equations that work for HORIZONTAL (x-direction) motion ALSO work for VERTICAL (y-direction) motion. • Simply substitute g for a and Δy for Δx • Generally, free falling objects start falling from rest, so it tends to simplify the equations by eliminating vi

The Big Misconception • The acceleration of gravity, g, is the same for all freefalling objects regardless of how long they have been falling, or whether they were initially dropped from rest or thrown up into the air. • BUT "Wouldn't an elephant free-fall faster than a mouse? " NO!! • WHY? • All objects free fall at the same rate of acceleration, regardless of their mass.

Position – Time Graphs • Recall that the slope of the line on a position versus time graph is equal to the velocity of the object. • This is really easy to calculate for objects moving at a constant velocity (i. e. having NO acceleration)

Position – Time Graphs • However, for objects WITH accelerated motion, the graphs look quite different: Changing Velocity Positive Changing Velocity Slow to Fast, Negative Changing Velocity Fast to Slow, Negative

Acceleration Graphs • Calculating the slope of those curved lines requires calculus, so we won’t be doing that…so, how do we determine the value of acceleration from a graph? • We use VELOCITY-TIME graphs! • The slope of the line on a velocity-time graph is acceleration: vf – vi tf - t i

Velocity – Time Graphs Positive Velocity Zero Acceleration

Velocity – Time Graphs Check Your Understanding The velocity-time graph for a two-stage rocket is shown below. Use the graph and your understanding of slope calculations to determine the acceleration of the rocket during the identified time intervals.

Ticker Tape Diagrams • A common way of analyzing the motion of objects in physics labs is to perform a ticker tape analysis. • A long paper tape is attached to a moving object and threaded through a device that places a mark on the tape at regular intervals of time - say every 0. 10 second. • As the object moves, it drags the tape through the "ticker, " thus leaving a trail of dots. The trail of dots provides a history of the object's motion and therefore a representation of the object's motion.

Ticker Tape Diagrams

Ticker Tape Diagrams Check Your Understanding: Imagine a car with a leaky engine that drips oil at a regular rate. As the car travels through town, it would leave a trace of oil on the street that would reveal information about the motion of the car. Analyze three traces of Renatta Oyle's car as shown below. 1. 2. 3.