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MOTION IN ONE DIMENSION • DYNAMICS – the study of motion and of physical concepts (such as forces & mass) • KYNEMATICS – the part of dynamics that describe motion MOTION IN ONE DIMENSION • Any motion involve the concept of: - displacement - velocity - acceleration • In this chapter we use these concepts to study the motion of objects undergoing constant acceleration DISPLACEMENT • Motion involve the displacement of an object from one place in space and time to another • The DISPLACEMENT Δx of an object is defined as its change in position, and is given by: Δx = xf - xi SI unit: m (Δ- denote any change of quantity) • VECTOR quantity - is characterized by having magnitude (size) and a direction • SCALAR quantity – is characterized by having magnitude but not direction • DISPLACEMENT – is a vector , because has both, magnitude and direction VELOCITY • SPEED → Scalar → magnitude • VELOCITY →vector → magnitude & direction • The average speed of an object over a given time interval is defined as the total distance traveled divided by the total time elapsed: v = d/t SI unit: m/s AVERAGE VELOCITY • The average velocity v during a time interval Δt is the displacement Δx divided by Δt v = Δx / Δt = (vf-vi) / ( tf -ti) SI unit: m/s • The velocity can be + or - ; depending on the sign of the displacement GRAPHICAL INTERPRETATION OF VELOCITY • Fig 1 • If a car move along the x-axis from A to B, the graph – straight line, because the car is moving with a ct. velocity, the same displacement Δx occurs in each time Δt • Fig 2 if is not a straight line, because velocity of the car is changing INSTANT VELOCITY • The average velocity doesn’t take into account the details of what happened during an interval of time • The instant velocity v – is the limit of the average velocity as the time interval Δt becomes infinitesimally small v ≈lim Δt→ 0 (Δx / Δt) SI unit: m/s • The SLOPE of the tangent line to the position vs. time curve at a “ given time” is defined to be the instantaneous velocity at that time (fig 3) • The instantaneous speed of an object, which is a scalar quantity, is defined as the magnitude of the instantaneous velocity ACCELERATION • The velocity of a car increase harder when you step harder on the gas pedal, and decrees when you apply the brake • The changing of an object’s velocity with time is called acceleration • The average acceleration a during the time int. Δt is the change of velocity Δv divided by Δt a = Δv/ Δt= (vf- vi)/ tf-ti) SI unit: m/s2 • An average acceleration of 5 m/s 2 means on average, the car increase its velocity by 5m/s every second in the + direction • When the object velocity and acceleration are in the same direction, the speed of the object increases with time • When the object’s velocity and acceleration are in opposite directions, the speed of the object decreases in time INSTANT ACCELERATION • Fig 4 • The instantaneous acceleration a is the limit of the average acceleration as the time interval Δt goes to zero a ≈lim Δt→0 (Δv/ Δt) SI unit: m/s2 • The instantaneous acceleration of an object at a given time = the SLOPE of the tangent to the velocity vs. time graph at that time MOTION DIAGRAM • Velocity and acceleration are sometimes confused with each other • A motion diagram is a representative of a moving object at successive intervals (fig5) a) the car moves the same distance in each interval. The car moves with ct. + velocity and has zero acceleration b) the velocity vector increases with time. The car is moving with a + velocity and a ct. +acceleration c) The car slows. The car is moving with + velocity but with a - acceleration ONE DIMENSIONAL MOTION WITH CT. ACCELERATION • When an object moves with ct. acceleration, the instantaneous acceleration at any point in a time interval is equal to the value of the average acceleration over the entire time interval (fig6) • a=a a =(vf -vi)/ tf -ti) • If ti =0; tf =t; vi =v0; vf =v • a= (v- v0)/t or • v = v0 + at velocity as a function of time • a steadily changes the initial velocity v0 by an amount at (fig7) • Because the velocity is increasing or decreasing uniformly with time, we can express the average velocity as the arithmetic average • v = (v0 + v)/2 • Δx= v t= [(v0 + v)/2] t • Δx= ½ (v0 + v)t for a= ct. • Δx= ½ (v0+v+at) t • Δx= v0 t + ½ at2 displacement as a function of time (for ct. a) • The area under the graph of v versus t for any object is = to the displacement Δx of the object • Δx= ½(v+v0)(v-v0)/a= (v2v02)/2a • v2= v02+ 2aΔx velocity as a function of displacement (for ct. a) FREE FALLING OBJECT • When air resistance is negligible, all objects dropped under influence of gravity near Earth's surface, fall towards Earth with the same ct. acceleration (law discovered by Galileo Galilei) • FREE FALLING object – any object moving freely under the influence of gravity alone, regardless of its initial motion • gEarth =9.8 m/s2 (for quick estimates g≈10 m/s2 )