Phy 211 General Physics I Chapter 2 Motion along a Straight Line Lecture Notes Galileo Galilei 1564 1642 Credited with establishing the scientific method Based his scientific hypotheses on observation and experimentation First to use telescope for astronomical observation Observed the following The craters and features on the Moon The moons of Jupiter the rings of Saturn Sun spots Based on his observations supported Copernican Theory Conclusively refuted Aristotelian ideology and contradicted Church doctrine Placed under house arrest as punishment Studied accelerated motion and established the first equations of kinematics Proposed the Law of Inertia which later became known as Newton s 1st law of Motion Displacement Motion An object is in motion if it changes its position relative to a reference point Origin Motion is relative r Position r a vector that connects an object s location to a reference point origin As an object moves the its position changes r Displacement Dr is the difference between the final position and the initial position of a moving object its magnitude is measured in meters m Origin 0 0 r2 position vector at t2 1 r1 position vector at t1 2 r Displacement Speed Velocity Velocity is a vector that reflects the time rate of change of displacement The direction of the velocity vector is the same as the r displacement vector The magnitude of a velocity vector is the Dr speed Both velocity and speed are measured in m s SI units Origin 0 0 r2 r1 1 v velocity r Displacement r r r displacement D r Average Velocity v avg time Dt Average Speed v v distance traveled avg avg Dt r r r Dr dr Instantaneous Velocity v lim Dt 0 Dt dt Speed v vr the magnitude of the velocity 2 Acceleration When the velocity vector is changing in time the acceleration vector describes the time rate of change of the velocity vector Acceleration a is a vector with the same direction as the change in velocity v vector It is measured in m s2 r r r change in velocity Dv Average Acceleration aavg time Dt r r 2r r Dv dv d r a lim 2 Instantaneous Acceleration Dt 0 Dt dt dt Example 1 A car accelerates from 0 to 30 m s in 6 s Find the average r acceleration r Dv 30 m s m a 5 avg Since v 30 m s Dt 6 s s2 Example 2 A ball descends in water with a velocity defined by v t 1t2 10t 5 assume SI units Find the instantaneous acceleration after 2 seconds r r dv d m a 1t 2 10t 5 2t 10 6 2 dt dt s Analyzing 1 D Motion How are the definitions of displacement velocity and acceleration applied to describe linear motion Example 1 A ball rolls down an incline from rest at to 0s at an acceleration of 2 5 m s2 r 1 What is the velocity after the rock has fallen 3 s r r r r r r Dv a aavg Dv v vo at eqn a Dt r m since vo 0 s the velocity down the incline is r r r m m v at 2 5 s2 3s 7 5 s down the incline a 2 How far does the rock travel during this 3 s There are 2 ways to approach this problem i Using r acceleration v r dv r r r r t r r r a dv adt dv adt v v o at or vo 0 r dt r t r r r r r r tr r dr r v vo at dr vdt dr vdt v o at dt dt ro 0 0 r 2 r r r r r 2 1 r ro vo t 2 at eqn b r ro 12 2 5 sm2 3s 11 25 m Analyzing 1 D Motion cont ii 3 Using initial and final velocity r r r r r r r r r vo v vo v Dr vavg Dr r ro t Dt 2 2 r r 0 ms 7 5 ms r ro 3s 11 25 m 2 eqn c How fast is the rock moving after it has fallen 10 m In this problem time is not explicitly given so we have to identify a workaround v v v vo at t o a r ro v ot 12 at 2 2 2 v vo 1 v vo v vo 1 v v o r ro vo a r r v 2 a o o 2 a a a a vvo v2o v2 v2o vv o v2o v2 2vv o v2o r ro r ro a 2a a 2a or v2 v2o 2a r ro v eqn d v2o 2a r ro 2 2 5 sm2 10m 7 07 ms Equations of Kinematics r To summarize the previous 2 slides when to 0 and a is constant a useful set of equations of motion has been derived for 1 D motion in scalar form a b c d v1 vo at r1 ro vot at2 r1 ro vo v1 t v12 vo2 2a r ro Velocity Equation Displacement Equation Average Velocity Galileo s Equation These equations will allow us to completely describe the motion of a moving object and represent our toolbox for studying kinematics Free Falling Bodies 1 2 3 4 The 1 D motion of objects subject to gravity The only motion considered is vertical use y for position There is no air resistance Acceleration of falling object is constant r r ay g y 9 8 m s2 y ay Example 1 Consider a rock dropped from rest 1 What is the velocity after the rock has fallen 3 s 2 How far does the rock travel during this 3 s 3 How fast is the rock moving after it has fallen 10 m Example 2 Consider a rock tossed into the air with an initial upward velocity of 5 m s 1 How long would it take for the rock to return to the thrower initial height 2 What is the highest position the rock reaches during its ascent Graphical Analysis of Velocity Acceleration position vs time graph The slope is the velocity velocity vs time graph The slope is the acceleration The area under the curve is the displacement
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