Motion General view: Deriving of motion equations in 3-Dimentional. Special view: Some special example of motions.

Mechanics It is an important branch of physics and deals with the effect of force on bodies. It is further divided into two parts 1. Kinematics : it deals with the study of motion of bodies Without any reference to the cause of motion. 2. Dynamics : In dynamics we discuss the motion of bodies under the action of forces.

MOTION 1. what is motion? when a body is continuously changing its position with respect to the surroundings (Frames of Reference), then we say that the body is in motion.

EXAMPLE : 1.When an athlete is running on the ground then he is continuously changing his position with respect to the audience who are sitting at rest. 2.We are continuously changing our position since morning till night with respect to earth which is at rest. 3.The earth is continuously changing position with respect to sun which is at rest.

Frames of Reference The object or point from which movement is determined Movement is relative to an object that appears stationary Earth is the most common frame of reference

system of coordinates Usually we show reference frame with coordinates system. In three dimensions this is as: x y z 1 i j 1 k 1 P Ex: P = (-1.8, -1.4, 1.2) so the vector -1.8 i – 1.4 j k will extend from the origin to P.

General view of motion, 3-D. Definitions Position ( p ) Position Vector ( r Trajectory Distance (d Displacement (  r )

Position ( p ) – where you are located start stop

Position Vector ( r )-An arrow from the center of coordinate to the position of body x z start stop riri rfrf y

start stop Trajectory – The path you drive (the path introduce from body movement) Distance (d ) – how far you have traveled, regardless of direction ( the length of trajectory ) It is a scalar.

Displacement (  r ) –Chang in position of an object, ( An arrow from initial point to final point ) start stop rr riri rfrf rr

x y z start stop riri rfrf rr

Distance vs. Displacement You drive the path, and your odometer goes up by 8 miles (your distance). Your displacement is the shorter directed distance from start to stop (green arrow). What if you drove in a circle? start stop

SPEED Defined as – How fast you go Average speed = distance travelled in a certain time s = distance / time = d / t Distance usually measured in metres; time in seconds; speed in metres per second (m / s) Tells us nothing about the direction travelled in It is usually an “average” value (unless measured and recorded every second!)

speed مقدار تندی متوسط به مبدأ و مقصد وابسته است مکانزمانمسافت طی شده تندی متوسط 1t1t1 d 11 =0 ? 2t2t2 d 21 d 21 /(t 2 - t 1 ) 3t3t3 d 31 d 31 /(t 3 - t 1 )

Speed تندی لحظه ای مقداری است که سرعت سنج اتومیبل در هر لحظه نشان می دهد تندی لحظه ای را می توان حد تندی متوسط در نظر گرفت طوریکه مبدأ و مقصد خیلی به هم نزدیک باشند S = Lim d fi t f - t i f i

Velocity Velocity (v) – how fast and which way; the rate at which position changes سرعت متوسط نسبت تغییربردارمکان ذره (بردارجابجایی ذره) بر زمان لازم برای این تغییر مکان (جابجایی) است V fi = ∆r∆r t f - t i = r f - r i سرعت کمیت برداری است چون حاصل تقسیم بردار جابجایی بر عدد(زمان جابجایی) است

Velocity راستای سرعت متوسط با راستای بردار جابجایی یکی است هم مقدار و هم راستای سرعت متوسط وابسته به مبدا ومقصد است r 31 r 41 2 r 21

Velocity –سرعت لحظه ای همان تندی لحظه ای است که راستای حرکت در آن مشخص است سرعت لحظه ای را می توان حد سرعت متوسط در نظر گرفت طوریکه مبدأ و مقصد خیلی به هم نزدیک باشند r 31 r 41 2 r 21 در شکل مشخص است که هر چه مقصد به مبدا نزدیکتر باشد بردار جابجایی به خط مماس بر مسیر نزدیکتر می شود لذا سرعت لحظه ای بر مسیر حرکت در هر لحظه مماس است در شکل بردارهای مماس در نقاط 2 و3 Instantaneous velocity is always tangential to trajectory V2V2 V3V3 V = Lim t f - t i r f - r i f i

Velocity از تعریف حد در ریاضیات نتیجه می شود که سرعت لحظه ای مشتق بردار مکان نسبت به زمان است. V = dt dr 1.از معادله بالا با دانستن تابعیت زمانی بردار مکان r(t) می توان سرعت را بدست آورد. اگر در مساله ای تابعیت زمانی سرعت را بدانیم میتوانیم به صورت زیر تابعیت زمانی مکان را بدست آوریم البته چون مساله تبدیل به یک مساله انتگرالی می شود باید درلحظه ای مثل t 1 مکان r 1 (t 1 ) معین باشد dr = V × dt r(t) – r 1 (t 1 ) = ∫ V × dt t1t1 t r(t) = ∫ V × dt + r 1 (t 1 ) t1t1 t 2. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

Speed vs. Velocity Average speed and the magnitude of average velocity are generally not equivalent because total distance and the absolute value of total displacement are generally not the same. The scalar absolute value (magnitude) of velocity is speed Average velocity magnitude is always smaller than or equal to average speed of a given particle

Example: Motion with constant velocity Constant velocity means 1.Direction of velocity is constant. 2.Magnitude of velocity is constant. Constant direction means: The motion is 1-D. Constant magnitude means: Then : X = V t + X 0, متحرکی موقع رفت نصف فاصله مسیری را با سرعت ثابت V 1 و نصف باقیمانده را با سرعت ثابت V 2 طی می کند موقع برگشت نصف زمان برگشت را با سرعت V 1 ونصف بقیه را با سرعت V 2 طی می کند. 1.سرعت متوسط موقع رفت؟ 2.سرعت متوسط موقع برگشت؟ 3.سرعت متوسط در رفت و برگشت؟ V = V

Acceleration The rate of change of velocity is acceleration – how an object's speed or direction changes over time, and how it is changing at a particular point in time. Average acceleration: The average acceleration is the ratio between the change in velocity and the time interval.

Acceleration Instantaneous acceleration (the acceleration at an instant of time) is defined as the limiting value of average acceleration as Δt becomes smaller and smaller. Under such a limit, a → a. If we know velocity as a function of time V(t), we can calculate acceleration as a function of time.

Integral relations of motion The above definitions can be inverted by mathematical integration to find:

Some special view: Kinematics of constant acceleration When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, with t 0 =0

Additional relations in constant acceleration motion V = V + V 0 2 Drive these equations from equations of previous page

مساله

Example: One dimensional motion If in start time of motion, the direction of initial velocity is the same as the direction of acceleration, then we have a motion in straight line. That is because v f = v 0 + a t So the direction of V f is the same as V 0 and the direction of motion doesn’t change. If we show this direction by X then we have: v f = v 0 + a t v avg = (v 0 + v f ) / 2  x = v 0 t + ½ a t 2 v f 2 – v 0 2 = 2 a  x

Acceleration due to Gravity 9.8 m/s 2 Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance). a = -g = -9.8 m/s 2 This acceleration vector is the same on the way up, at the top, and on the way down! ground y v f = v 0 - g t v avg = (v 0 + v f ) / 2  y = - ½ g t 2 + v 0 t v f 2 – v 0 2 = -2 g  y

Velocity & Acceleration Sign Chart V E L O C I T Y ACCELERATION ACCELERATION Moving forward; Speeding up Moving backward; Slowing down - Moving forward; Slowing down Moving backward; Speeding up

Sample Problems 1.You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration? 2.A brick is dropped from 100 m up. Find its impact velocity and air time. 3.An arrow is shot straight up from a pit 12 m below ground at 38 m/s. a.Find its max height above ground. b.At what times is it at ground level?

Multi-step Problems 1.How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m? 2.A dune buggy accelerates uniformly at 1.5 m/s 2 from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled m/s m Answer:

Graphing ! x t A B C A … Starts at home (origin) and goes forward slowly B … Not moving (position remains constant as time progresses) C … Turns around and goes in the other direction quickly, passing up home 1 – D Motion

Graphing w/ Acceleration x A … Start from rest south of home; increase speed gradually B … Pass home; gradually slow to a stop (still moving north) C … Turn around; gradually speed back up again heading south D … Continue heading south; gradually slow to a stop near the starting point t A B C D

Tangent Lines t SLOPEVELOCITY Positive Negative Zero SLOPESPEED SteepFast GentleSlow FlatZero x On a position vs. time graph:

Increasing & Decreasing t x Increasing Decreasing On a position vs. time graph: Increasing means moving forward (positive direction). Decreasing means moving backwards (negative direction).

Concavity t x On a position vs. time graph: Concave up means positive acceleration. Concave down means negative acceleration.

Special Points t x P Q R Inflection Pt. P, RChange of concavity Peak or ValleyQTurning point Time Axis Intercept P, S Times when you are at “home” S

Curve Summary t x A B C D

All 3 Graphs t x v t a t

Graphing Tips Line up the graphs vertically. Draw vertical dashed lines at special points except intercepts. Map the slopes of the position graph onto the velocity graph. A red peak or valley means a blue time intercept. t x v t

Graphing Tips The same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes! Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis. a t v t

Area under a velocity graph v t “forward area” “backward area” Area above the time axis = forward (positive) displacement. Area below the time axis = backward (negative) displacement. Net area (above - below) = net displacement. Total area (above + below) = total distance traveled.

Area The areas above and below are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this too. v t “forward area” “backward area” t x

Area units Imagine approximating the area under the curve with very thin rectangles. Each has area of height  width. The height is in m/s; width is in seconds. Therefore, area is in meters! v (m/s) t (s) 12 m/s 0.5 s 12 The rectangles under the time axis have negative heights, corresponding to negative displacement.

Graphs of a ball thrown straight up x v a The ball is thrown from the ground, and it lands on a ledge. The position graph is parabolic. The ball peaks at the parabola’s vertex. The v graph has a slope of -9.8 m/s 2. Map out the slopes! There is more “positive area” than negative on the v graph. t t t