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LECTURE 3 1D MOTION Instructor: Kazumi Tolich Lecture 3 2 ¨ Reading chapter 2-1 to 2-6 ¤ One dimensional motion n Position, displacement, and distance n Velocity and speed n Acceleration n Motion with a constant acceleration Vector and scalar quantities 3 ¨ A vector quantity has a magnitude and direction. ¤ ¤ ¤ ¤ ¨ Displacement (m): how far something moved in what direction Velocity (m/s): how fast something is moving in what direction Acceleration (m/s2): how fast velocity is changing in what direction etc. A scalar quantity has only magnitude. ¤ ¤ ¤ ¤ Time (s): how long it has been. Temperature (K): how hot something is. Mass (kg): how much stuff there is. etc. Coordinate system and position 4 A coordinate system defines the position of an object. ¨ You need to define where the origin is, and which direction is the positive direction. ¨ x0 Initial (x0) and final (xf) Positions of person Displacement and distance 5 ¨ ¨ ¨ ¨ ¨ Displacement is the difference in the initial and final positions: Δx = xf – x0 (in the x direction). In calculating the displacement of an object, how it traveled from the initial to the final position does not matter. Displacement is a vector quantity. Total distance traveled is total length of travel. Distance is a scalar quantity. Clicker question: 1 6 Clicker question: 2 7 Elapsed time 8 ¨ ¨ ¨ Time is change, or the interval over which change occurs. The SI unit for time is the second, s. Elapsed time Δt is Δt = tf − t0 where tf is the final time, and t0 is the initial time. ¨ If we define t0 = 0, Δt = tf ≡ t Average velocity and average speed 9 ¨ The average velocity (in the x direction) is defined to be v≡ ¨ Δx xf − x0 = Δt tf − t0 The average speed of the trip is defined to be average speed ≡ ¨ total distance Δt Average velocity is a vector quantity, and average speed is a scalar quantity. Clicker question: 3 10 Instantaneous velocity and speed 11 ¨ Instantaneous velocity in the x direction is defined to be Δx v ≡ lim Δt →0 Δt ¨ ¨ Instantaneous velocity is a vector quantity. Instantaneous speed is the magnitude of the instantaneous velocity. Instantaneous vs. average speed 12 ¨ Why is instantaneous speed more interesting to the highway patrol than average speed? ¤ Suppose you drive 100 km in one hour. Your average speed would be 100 km/h. ¤ This is your average speed over the whole trip. ¤ But how likely is it that you were traveling at exactly 100 km/h the whole time? Average and instantaneous accelerations 13 ¨ Acceleration is the rate of change of velocity, or how quickly velocity is changing. Average acceleration in the x direction is defined to be ¨ Δv vf − v0 = Δt tf − t0 Instantaneous acceleration is defined to be ¨ a≡ Δv Δt →0 Δt a ( t ) ≡ lim ¨ ¨ Acceleration is a vector quantity. It points in the direction of Δv. When acceleration is constant, the instantaneous and average acceleration are the same. Directions of a and v 14 ¨ ¨ ¨ When the direction of velocity and acceleration are the same, the object is speeding up. When the direction of velocity and acceleration are opposite, the object is slowing down. When an object slows down, its acceleration is opposite to the direction of its motion. This is known as deceleration. Acceleration and force 15 Acceleration is caused by force. ¨ The direction of acceleration of an object is the same as the force applied to accelerate it. ¨ Clicker question: 4 16 Motion with constant acceleration 17 ¨ For an object with an initial position, x0, initial velocity, v0, and a constant acceleration, a, ¤ the velocity, v, as a function of time, t, is given by v = v0 + at ¤ the position, x, as a function of time, is given by x = x0 + v0t + 12 at 2 ¤ The velocity as a function of displacement, Δx, is given by v 2 = v02 + 2a ( x − x0 ) = v02 + 2aΔx Demo: 1 18 ¨ Incline with Flash Lights ¤ Demonstration of the distance and the velocity formula under a constant acceleration. 1 2 x ( t ) = x0 + v 0 t + at 2 v ( t ) = v0 + at Example: 1 19 ¨ A motor cycle is moving at v0 = 30.0m/s when the rider applies the brakes, giving the motorcycle a constant deceleration. At t1 = 3.0s, after braking begins, the speed decreases to v1 = 15.0m/s. What distance does the motorcycle travel from the instant braking begins until it comes to rest? Example: 2 20 ¨ At t = 0 a ball, initially at rest, starts to roll down a ramp with constant acceleration. Suppose it moves 1 foot between t = 0 s and t = 1 sec. How far does it move between t = 1sec and t = 2 sec? ? 4ft 9ft 16ft 1ft