Ch 2. Describing Motion: Kinematics in 1-D.

Ch 2. Describing Motion: Kinematics in 1-D. Introduction Kinematic Equations are mathematic equations that describe the behavior of an object in terms of its motion as a function of time. Kinematics is the physics of Galileo! (1564-1642) Galileo Galilei (15 February 1564 8 January 1642) Motion in physics is broken down into 3 basic categories a.) Translational Motion - Straight line motion b.) Rotational Motion - Spinning motion often use polar coordinates to solve these problems c.) Vibrational Motion - Involves a back-and-forth periodicity. Diatomic Molecule Moving Through Space Oxygen O 2, Hydrogen H 2, Nitrogen N 2, Fluorine F 2, Chlorine Cl 2 Vibrational Rotational Translational 1

Lesson 1. Speed, Displacement, Velocity 1. Speed In general describes how fast an object is moving regardless of what direction it is moving. Average speed is how fast an object is moving, on average, over an entire trip. v ave. d total t total = speed Example: An athletic ant starts at the 2 mile mark and it takes him 1 hr. to get to the origin and.75 hrs. to go from the origin back to 1 mile mark, where he stops. What was his average speed for this entire trip? y (miles) 1 mile 2 mile x (miles) 2

Displacement and Average Velocity Displacement - Change in position (distance with direction) To be consistent and to have a point of reference from which to specify a direction when taking a change in position one must specify a coordinate system. x = x f x i Ex. A particle starts at position xi = +7 m and finishes at xf = -1 m. Find x Average Velocity - How fast an object is moving specifically in a certain direction. v = x t = x f x i t f t i Ex.: A particle starts at position xi = -7 m and finishes at xf = -2m and it takes 2 sec from start to finish. Find x, t, v Velocity is PATH INDEPENDENT (Only initial and final positions matter!) Speed is PATH DEPENDENT (total distance traveled matters!) 3

Example: What if a bumble bee starts at the 2 mile mark and it takes her1 hr. to get to the origin following the zig-zag path shown which has a total distance of 3 miles. It then takes her.75 hrs. to go from the origin back to the1 mile mark along the 1.5 mile zig-zag path shown. a.) What was her average speed for this entire trip? b.) What was her displacement? c.) What was her average velocity? y (miles) 1 mile 2 mile x (miles) 4

Lesson 1 HW Problems: Speed, Displacement, and Average Velocity 1. Consider a deer that runs from point A to point B. The distance the deer runs can be greater than the magnitude of its displacement, but the magnitude of the displacement can never be greater than the distance it runs. (Answer and Explain Why) A) True B) False 2. Consider a car that travels between points A and B. The car's average speed can be greater than the magnitude of its average velocity, but the magnitude of its average velocity can never be greater than its average speed. (Answer and Explain Why) A) True B) False 3. What must be your average speed in order to travel 23 km in 3.25 hrs? (7.8 km/hr) 4. You are driving home from school steadily at 65 mph for 13 miles. It then begins to rain and you slow to 55 mph. You arrive home after driving 3 hrs and 2 minutes*. (a) How far is your hometown from school? (b) What was your average speed? *Note: convert minutes to hours. (23.333 mi, 61 mi/hr) 5. A person jogs 8 complete laps around a quarter-mile track in a total time of 12.5 minutes. Calculate the average speed and average velocity in m/s? (4.3 m/s, m/s) 5

6. A horse canters away from its trainer in a straight line, moving 13m away in 14 sec. It then turns abruptly and gallops halfway back in 4.8 sec. Calculate (a) its average speed and (b) its average velocity for the entire trip, using away from the trainer as the positive direction. (1.4 m/s, 3.46 m/s) 7. An airplane travels 21km at a speed of 8km/h and then encounters a tailwind that boosts its speed to 1km/hr for the next 18km. What was the total time for the trip? What was the average speed of the plane for this trip? (4.425 hrs, 881 km/hr) 8. A bowling ball traveling with a constant speed hits the pin at the end of a bowing lane 16.5 m long. The bowler hears the sound of the ball hitting the pins 2.5 sec after the ball is released from his hands. What is the speed of the ball? The speed of sound is 34 m/s. (6.73 m/s) 6

Position (m) (x-coord.) Lesson 2: Position and Velocity Analysis Position vs Time Graph x Y 1 Y 2 Time (s) Position (m) Position (m) 1 1 5 2 4 1 3 9 15 4 16 2 5 25 25 Position vs. Time 3 25 2 Movement 2 15 1 Movement 1 5 1 2 3 4 5 6 Time (sec) 7

Position vs. Time Graph for a Complete Trip 3 25 2 B C 15 D Position (m) 1 5 A F -5-1 -15 1 2 3 4 5 6 Time (s) E x y Time (s) Position (m) 1 2 2 2 25 15 45-1 6 Find the average speed and average velocity as the object moves from time (A to C), and (C to F), Find the average speed and average velocity as the object goes from time A to F. 8

Velocity vs Time (Constant Velocity) x(m) 8 7 6 5 4 3 2 1 Position Function 1 2 3 4 5 x vave t t x t(sec) Area! 4 Velocity Function v (m/s) 3 2 1 x 1 2 3 4 5 t(sec) 9

Velocity vs. Time Graph for a Complete Trip 3 25 2 B C 15 D Position (m) 1 5 A F -5-1 -15 1 2 3 4 5 6 Time (s) E Velocity (m/s) 25 2 15 1 5-5 -1-15 -2-25 A B E F B C C D E D 1 2 3 4 5 6 Time (s) 1

Instantaneous Velocity Recall: v ave x x t f t t xt f t i i x (Average velocity) Consider the function x(t): xt 3 m + 1 m s 2 t 2 -.5 m s 4 t 4 A. B. v v ave ave 525. m - 35. m 3 s - 2 s 39. 69 m - 35. m 2.2 s - 2 s m 175. s x 2 s < t < 3 s t = 1 sec m 2344. s x 2 s < t < 2.2 s t =.2 sec 6 5 slope = 17.5 m/s 6 5 slope = 23.45 m/s 4 4 x(m) 3 2 x(m) 3 2 1 1 1 2 3 4 t(sec) 1 2 3 4 t(sec) t (sec) vave (m/s) 1 17.5.2 23.45.1 23.98.1 23.998 Define: The instantaneous velocity at time ti is the slope of the line tangent to the curve X(t) at the time ti. v t dx t dt lim X t f X t i t f t i tf ti 11

Lesson 2 HW: Position and Velocity Graphs 1. Graph the velocity vs time graph using the position vs time graph below Velocity vs Time Velocity (m/s) Position (m) 8 6 4 Position vs Time 2-2 -4-6 1 2 3 4 Time (s) 1 2 3 4 Time (s) Calculate the distance, displacement, average speed and average velocity over the time interval from 1 to 3 sec. 12

Position (m) Velocity (m/s) 2. Graph the position vs time plot below. Assume the particle starts at position x = 2 m. Velocity vs Time 8 6 4 2-2 -4-6 1 2 3 4 Time (s) Position vs Time 1 2 3 4 Time (s) Using the velocity graph describe whether the particle is moving forward (+x) or backward (-x) and whether it is speeding up, slowing down, or moving at constant speed over each of the intervals. to 1 sec 1 to 2 sec 2 to 3 sec 3 to 4 sec 13

Velocity (m/s) Position (m) 3. Graph the velocity vs time graph using the position vs time graph below. Position vs Time 5 4 3 2 1-1 -2-3 -4-5 1 2 3 4 Time (s) Velocity vs Time 1 2 3 4 Time (s) Calculate the distance, displacement, average speed and average velocity over the time interval from 1 to 4 sec. 14

Position (m) Velocity (m/s) 4. Graph the position vs time plot below. Assume the particle starts at position x = m. Velocity vs Time 8 6 4 2-2 -4-6 -8 1 2 3 4 Time (s) Position vs Time 1 2 3 4 Time (s) Using the velocity graph describe whether the particle is moving forward (+x) or backward (-x) and whether it is speeding up, slowing down, or moving at constant speed over each of the intervals. to 1 sec 1 to 2 sec 2 to 3 sec 3 to 4 sec 15

5. A position-time graph for a particle moving along the x axis is shown below. (a) Find the average velocity in the time interval t = 1.5 s to t = 4. s. (b) Determine the instantaneous velocity at t = 2. s by measuring the slope of the tangent line shown in the graph. (c) At what value of t is the velocity zero? (-2.4 m/s,-3.7m/s, 4 sec) 16

Lesson 3: Acceleration When the instantaneous velocity of a particle is changing with time, the particle is accelerating a ave = v t = v f v i t f t i Units: a m/s m s s ave 2 Example: A particle is being accelerated at + 1 m/s 2 in the x-direction for both parts a) and b) below. a.) If the particle starts with a velocity of + 15 m/s how long does it take the particle to have a velocity of + 25 m/s? b.) If the particle starts with a velocity of 15 m/s how long does it take the particle to have a velocity of + 25 m/s? 17

Positive and Negative Accelerations You must look at the initial velocity direction (+ or ) and the sign (+ or ) of the acceleration to see if the acceleration in a given problem means speeding up (acceleration) or slowing down (deceleration). If the signs for the velocity and the acceleration are the same then it is speeding up. If the signs are opposite then it is slowing down. Note: Take to the left as NEGATIVE direction and to the right as POSITIVE Speeding Up =Acceleration (a = 1 m/s 2, v i = 5 m/s, Δt = 1 sec ) Final Initial Slowing Down = Deceleration (a = 1 m/s 2, v i = + 15 m/s, Δt = 1 sec ) Initial Final 18

Acceleration (m/s 2 ) Velociy (m/s) Velocity to Acceleration Graphs a ave = v t = rise run = slope of v vs. t graph Velocity vs Time 18 16 14 12 1 8 6 4 2.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) Acceleration vs Time.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) 19

Velocity (m/s) Acceleration (m/s 2 ) Acceleration to Velocity Graphs v = t a = Base Height = Area under Accel Graph 14 12 1 8 6 4 2-2 -4 Acceleration vs Time -6.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) Note: vi = 8 m/s at time t = Velocity vs Time.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) 2

Lesson 3 HW: Acceleration 1. A 5.-g superball traveling at 25. m/s bounces off a brick wall and rebounds at 22. m/s. A high-speed camera records this event. If the ball is in contact with the wall for 3.5 ms, what is the magnitude of the average acceleration of the ball during this time interval? (Note: 1 ms = 1 3 s.) (13428 m/s) 2. A particle starts from rest and accelerates as shown in the figure below. Determine (a) the particle's speed at t = 1. s and at t = 2. s, and (b) the distance traveled in the first 2. s. (+2 m/s, +5 m/s, 262.5 m/s) 21

3. A velocity-time graph for an object moving along the x axis is shown in the figure below. (a) Plot a graph of the acceleration versus time. (b) Determine the average acceleration of the object in the time intervals t = 5. s to t = 15. s and t = to t = 2. s. (1.6 m/s 2,.8 m/s 2 ) 22

4. The figure below shows a graph of vx versus t for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line. (a) Find the average acceleration for the time interval t = to t = 6. s. (b) Estimate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant. (c) When is the acceleration zero? (d) Estimate the maximum negative value of the acceleration and the time at which it occurs. (1.33 m/s 2, 2 m/s 2, t=6 and t 1, about 1.5 m/s 2 ) 23

Lesson 4: Constant Acceleration a(m/s 2 ) We make the assumption that the acceleration does not change. Near the surface of the earth, (where most of us spend most of our time) the acceleration due to gravity is approximately constant a g = 9.8 m/s 2 a t i = t f = t t (s) Area! Slope! v f = v i + a t 1. v(m/s) v f v i t i = Area! t f = t t (s) Slope! x(m) x f x i t i = t f = t t (s) x x + v t f = i i + 1 2 a t2 2. 24

Solving for the 3 rd constant acceleration equation Solve equation 1 for t and substitute t into equation 2 to get the following equation. 2 2 v v 2 a x 3. f i 25

Throwing a Ball Vertically Straight Up Let g = 1 m/s 2 for this problem. Assume the ball is thrown upward at time t= with an initial speed of 4 m/s. (a) How long does the ball take to reach the top of the flight? (b) What speed does it have at time t = 1, 2, 3 and 4 seconds into flight? (c) What distance does it travel for consecutive seconds up to 4 seconds (i.e Δy =? from to 1sec, from 1 to 2sec, etc.). (d) What is the acceleration of the ball at the top of its flight? Understanding the Graphs for this problem 26

Lesson 4 HW: Constant Acceleration Equations 1. A truck covers 4. m in 8.5 s while smoothly slowing down to final speed 2.8 m/s. (a) Find its original speed. (b) Find its acceleration. (6.61 m/s), -.448 m/s 2 ) 2. An object moving with uniform acceleration has a velocity of 12. cm/s in the positive x direction when its x coordinate is 3. cm. If its x coordinate 2. s later is 5. cm, what is its acceleration? (-16 cm/s 2 ) 3. A speedboat moving at 3. m/s approaches a no-wake buoy marker 1 m ahead. The pilot slows the boat with a constant acceleration of 3.5 m/s 2 by reducing the throttle. (a) How long does it take the boat to reach the buoy? (b) What is the velocity of the boat when it reaches the buoy? (4.53 s, 14.1 m/s) 27

4. In the Daytona 5 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straightaway at 71.5 m/s. The driver of the Thunderbird realizes he must make a pit stop, and he smoothly slows to a stop over a distance of 25 m. He spends 5. s in the pit and then accelerates out, reaching his previous speed of 71.5 m/s after a distance of 35 m. At this point, how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed? (958 m) 5. A jet plane lands with a speed of 1 m/s and can accelerate at a maximum rate of 5. m/s 2 as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time interval needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is.8 km long? (2 sec, No, but explain why) 28

6. A car traveling at a constant speed of 3 m/s passes a trooper hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets in chase after the car with a constant acceleration of 3. m/s 2. How long does it take the trooper to overtake the speeding car? (21 sec) 7. A ball is thrown directly downward, with an initial speed of 8. m/s, from a height of 3. m. After what time interval does the ball strike the ground? (1.79 sec) 8. A student throws a set of keys vertically upward to her sorority sister, who is in a window 4. m above. The keys are caught 1.5 s later by the sister's outstretched hand. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught? (1 m/s, 4.68 m/s downward) 29

9. It is possible to shoot an arrow at a speed as high as 1 m/s. (a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up? (b) How long would the arrow be in the air? (51 m, 2.2 sec) 1. A freely falling object requires 1.5 s to travel the last 3. m before it hits the ground. From what height above the ground did it fall? (38.2 m) 3

Final Notes x(t) Slope v(t) Area Under Curve a(t) Problem Solving with the constant acceleration equations 1. Write down all three equations in the margin 2. a = 9.8 m/s 2 for free fall problems 3. Analyze the problem in terms of initial and final sections. 31