Chapter 2 Kinematics in one dimension
Galileo - the first modern kinematics 1) In a medium totally devoid of resistance all bodies will fall at the same speed 2) During equal intervals of time, a falling body receives equal increments of speed (constant acceleration) 1564-1642
I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing 10 times the other, if allowed to fall at the same instant from a height of 100 cubits, would so differ in speed that when the heavier had reached the ground the other would not have fallen more than 10 cubits. I, who have made the test can assure you that a cannon ball weighing one or two hundred pounds, or more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound. Salviati in Two New Sciences, 1636, by Galileo
Idealization and extrapolation - Totally devoid of resistance was not possible (nature abhors a vacuum), so Galileo made things worse: - If deviations from his law are far worse in a dense medium than a thin one, is it not reasonable to suppose that they would disappear altogether if the medium were absent?
Describing motion Complete description gives position as a function of time: x(t) e.g. x=(10m/s) t If t = 1 s, x = 10 m If t = 2 s, x = 20 m e.g. x = a t 2 =(10 m/s 2 ) t 2 If t = 1 s, x = 10 m If t = 2 s, x = 40 m
Describing motion Complete description gives position as a function of time: x(t) But speed and acceleration interesting too: Want v(t), a(t) Also v(x) Should be derivable from x(t)
1) Displacement Vector from initial to final position x + Δx = x 0 Δ x = x x 0
In 1d, the sign indicates direction: Δ x > 0, displacement to right Δ x < 0, displacement to left Δ x = 0, no displacement
3 s 0 s 1 s 2 s What is the total distance travelled? (odometer reading) A: 7 cm B: 22 cm C: 10 cm D: 1 cm E: -2 cm
3 s 0 s 1 s 2 s What is the final displacement? A: 7 cm B: 4 cm C: 10 cm D: 1 cm E: -2 cm
Graph the ball s position vs time 14 13 12 11 10 9 Position (cm) 8 7 6 5 4 3 2 1 0 0 1 2 3 4 Time (s)
2) Speed and Velocity Time rate of change of position Average speed = distance/time = s/ t (scalar)
If the first100 km of a 200 km trip are travelled at 50 km/h, and the second 100 km are travelled at 100 km/h, what is the average speed? A: 50 km/h B: 100 km/h C: 75 km/h D: 66 km/h E: 33 km/h
3 s 0 s 1 s 2 s What is the average speed? A: 2 cm/s B: -2 cm/s C: 28/3 cm/s D: 22/3 cm/s E: 13/3 cm/s
Average velocity = displacement/time (vector) v = Δ x Δt Units for speed or velocity: m/s
3 s 0 s 1 s 2 s What is the average velocity? A: 2 cm/s B: -2 cm/s C: 28/3 cm/s D: 22/3 cm/s E: -2/3 cm/s
Constant velocity: equal intervals of distance in equal intervals of time --> slope of a straight line graphical representation x vs t x t x v = Δx Δt t
Constant velocity: displacement is area under v vs t curve Δx = vδt = area graphical representation v vs t v v Δt t
Changing velocity: average speed depends on time interval -- curve on x vs t graph Here velocity increases in time x t
Instantaneous velocity v = lim Δt 0 Δ x Δt x t Represents tangent to x vs t curve.
For an arbitrary v(t), in a small interval, the displacement is the area (approx.): Δx i = v i Δt i v i v Δt i t
so the total displacement is the total area (if the intervals are small enough): Δx = Δx 1 + Δx 2 +... v t
In particular, if v vs t is a straight line (say, v = at): displacement is area of a triangle (here x = x - 0) x = 1 2 (base)(height) = 1 2 at 2 at v t
3) Acceleration Time rate of change of velocity Average acceleration! a = Δ! v Δt =! v v! 0 t t 0 Units: m/s 2
Constant acceleration: slope of a straight line (v vs t) For an object starting from rest, v = at v x t x Δv a = Δv Δt t
Changing acceleration: average acceleration depends on time interval -- curve on v vs t graph Here acceleration increases in time vx t
Instantaneous acceleration a = lim Δt 0 Δ v Δt vx t Represents tangent to v vs t curve.
displacement velocity (time rate of change of displacement) acceleration (time rate of change of velocity) jerk (time rate of change of acceleration) jounce (snap) (time rate of change of jerk) (crackle) (time rate of change of snap) (pop) (time rate of change of crackle)
Positive acceleration: acceleration in pos. dir n speeding up if v > 0 v t slowing down if v < 0 (deceleration) v t
Negative acceleration: acceleration in neg. dir n slowing down if v > 0 (deceleration) v t speeding up if v < 0 (acceleration) v t
(a) Suppose a NASCAR race car is moving to the right with a constant velocity of +82 m/s. What is the acceleration of the car at that instant? (b) Twelve seconds later, the car is halfway around the track and traveling in the opposite direction with the same speed. What is the average acceleration of the car during the 12 seconds? A: (a) 0, (b) 0 B: (a) 82 m/s 2, (b) 82 m/s 2 C: (a) 0, (b) 6.8 m/s 2 D: (a) 0, (b) -164 m/s 2 E: (a) 0, (b) -13.7 m/s 2
4) Kinematic equations (constant acceleration) Quantities: x, v, a, t (and their initial values, x 0, v 0, a 0, t 0 ) For constant acceleration, a = a 0 Usually choose coord system so x = x 0 = 0 at t = t 0 = 0 Leaves 5 quantities typically (v0 not zero in general) Want x(t), v(t), which give v(x), x(v,t)
Acceleration: a = v/ t = constant Velocity: a = v v 0 = v v 0 t t t 0 v = v 0 + at v v0 t
Position (displacement) evaluate area v Δx = v 0 t + 1 2 (v v 0)t v0 Δx = v 0 t + 1 2 at 2 t x = x 0 + v 0 t + 1 2 at 2 Taking x0 = 0, x = v 0 t + 1 2 at 2
v = v 0 + at 1 all but x x = v 0 t + 1 2 at 2 2 all but v Eliminate t from 1 and 2: v 2 = v 0 2 + 2ax 3 all but t Eliminate a from 1 and 2: x = 1 2 (v + v 0 )t = vt 4 all but a Eliminate v0 from 1 and 2: x = vt 1 2 at 2 5 all but v0
5) Applications Draw a picture use it to identify kinematic quantities Choose +ve direction and origin (can be for each object, or deal with x0) Identify quantities that are given and that are needed Identify the equations that relate them Solve
Example 2.28 (a) What is the magnitude of the average acceleration of a skier who, starting from rest, reaches a speed of 8.0 m/s when going down a slope for 5.0 s? (b) How far does the skier travel in this time?
Example A person is 11 m from a bus as it starts to leave the stop with an acceleration of 1.0 m/s 2. If the person runs after the bus at 5.0 m/s, how long does it take to catch the bus (assuming the driver has not taken Bob Newhart s driving course)?
Homework C&J 2.35 In a historical movie, two knights on horseback start from rest 88.0 m apart and ride directly toward each other to do battle. Sir George's acceleration has a magnitude of 0.300 m/s 2, while Sir Alfred's has a magnitude of 0.200 m/s 2. Relative to Sir George's starting point, where do the knights collide?
5) Free-fall Acceleration toward earth independent of mass, and (nearly) constant (neglecting air resistance) acceleration due to gravity: g = 9.8 m/s 2
A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone? y a v v o t? -9.80 m/s 2 0 m/s 3.00 s y = v 0 t + 1 2 at 2 = -44.1 m
Maximum height ymax a v vo? -g 0 m/s v0 ymax 2 0 v = 2g t
Speed of an object falling through h y a v v o t h g? 0 v 2 = 2gh v = 2gh
If you drop an object, in the absence of air resistance, it accelerates downward at 9.8 m/s 2. If instead you throw it downward, its downward acceleration after release is A. less than 9.8 m/s 2 B. 9.8 m/s 2 C. more than 9.8 m/s 2
A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown: A. upward B. downward C. neither -- they hit at the same speed
Symmetry What is the speed at the original position? v 2 v 2 0 = 2ay v 2 2 If y = 0, then = v 0 v = ±v 0
C&J 2.64 A roof tile falls from rest from the top of a building. An observer inside the building notices that it takes 0.20 s for the tile to pass her window, which has a height of 1.6 m. How far above the top of this window is the roof?
CJ 2.40 A Boeing 747 Jumbo Jet has a length of 59.7 m. The runway on which the plane lands intersects another runway. The width of the intersection is 25.0m. The plane decelerates through the intersection at a rate of 5.70 m/s2 and clears it with a final speed of 45.0 m/s. How much time is needed for the plane to clear the intersection?
6) Graphical Analysis
y y = v0t gt 1 2 t v = v0 gt a = g 2
A train car moves a long a straight track. The graph shows the position as a function of time for this train. The graph shows that the train: A. speeds up all the time. B. slows down all the time C. speeds up part of the time and slows down part of the time. D. moves a constant velocity.
C&J problem 2-45 The drawing shows a device that you can make with a piece of cardboard, which can be used to measure a person s reaction time. Hold the card and suddenly drop it. Ask a friend to try and catch the card between his or her thumb and index finger. Initially, your friend s fingers must be level with the asterisks at the bottom. By noting where your friend catches the card, you can determine his or her reaction time in milliseconds (ms). Calculate the distances d 1, d 2, and d 3.
Example: relative motion A ball is dropped from rest from an 8-m building at the same time another ball (directly below the first) is thrown up from the ground at 8 m/s. Where and when do they meet? When? a) 1 s b) 2 s c) 4 s d) 8 s e) they do not meet
The principle of equivalence The happiest thought of my life The gravitational field has only a relative existence... Because for an observer freely falling from the roof of a house - at least in his immediate surroundings - there exists no gravitational field. -- Einstein
h 8m y 1 = 1 2 gt 2 y 2 = v 0 t 1 2 gt 2 v0=8m/s t = 0s.1s.2s. 1.0 s y 1 + y 2 = h = 8m v 0 t = h = 8 m t = h v 0 = 8 m 8 m/s = 1 s
Viewed in a falling reference frame
Viewed in a falling reference frame
Homework C&J 2-47 Review Conceptual Example 15 before attempting this problem. Two identical pellet guns are fired simultaneously from the edge of a cliff. These guns impart an initial speed of 30 m/s to each pellet. Gun A is fired straight upward, with the pellet going up and then falling back down, eventually hitting the ground beneath the cliff. Gun B is fired straight downward. In the absence of air resistance, how long after pellet B hits the ground does pellet A hit the ground?