Definitions. Mechanics: The study of motion. Kinematics: The mathematical description of motion in 1-D and 2-D motion.

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Alaina McLaughlin
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1 Lecture 2

3 Definitions Mechanics: The study of motion. Kinematics: The mathematical description of motion in 1-D and 2-D motion. Dynamics: The study of the forces that cause motion.

4 Chapter Outline Consider in this chapter only motion in one dimension: motion along a straight line. First define position, displacement, velocity and acceleration. Then, using these concepts, study the motion of objects traveling in 1-D with a constant acceleration.

5 Definition of Motion Motion represents a continuous change in the position of an object. Example of motion: A car moving down a highway.

consider to be the origin of a coordinate system.

6 Particle s Position Particle s Position is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system. Positions to the right of the origin are positive. Positions to the left of the origin are negative.

7 Displacement Displacement of a particle x is defined as its change in position in some time interval. In other word; it is the difference between the final position x f, and the initial position x i. x = x f x i

8 Displacement A displacement to the right will be a positive displacement. For example starting with x i = 60 m and ending at x f = 150 m, the displacement is :

9 Displacement A displacement to the left will be a negative displacement. For example starting with x i = 150 m and ending at x f = 60 m, the displacement is :

10 Distance and Displacement Distance is the absolute value of the displacement. Distance is always positive and tells how far something is from something else but does not tell us whether it is to the right or to the left. Units are important in Physics (and in all of Science) In the lab, we will usually measure distance or displacement in units of meters (m). Distance or displacement could also be measured in centimeters (cm) or kilometers (km) or even miles (mi).

11 Example Find the displacement, and the distance between A & D

12 Ans. Displacement

13 Ans.

14 Ans. Total Distance = distance from A to B + distance from B to C + distance from B to C = 180 m m m = 420 m.

The average Speed of an object over a given time interval is defined as the total distance traveled divided by the total time

15 Speed & Velocity The average Velocity during a time interval t is the displacement Δx divided by the time t : θ avg = x t = x f x i t f t i SI Unit: meter per second (m/s) Velocity is a vector quantity. The average Speed of an object over a given time interval is defined as the total distance traveled divided by the total time elapsed: Average speed = total distance total time SI Unit: meter per second (m/s) Speed is a a scalar quantity. Speed & Velocity is also measured in km/h (and even in mi/hr).

16 Speed & Velocity question If you run from x=0 m to x=25 m and back to you starting point in a time interval of 5 sec. Find the average velocity and average speed Ans. : θ avg = x t = x f x i t f t i θ avg = = 0 m/sec The average velocity is zero meter/sec

17 Speed & Velocity Ans. : Average Speed = 25 m+25 m 5 = 50 5 = 10 m/sec The average speed is 10 meter/sec

18 Example The position versus time for a certain particle moving along the x axis is shown in Figure bellow. Find the average velocity in the time intervals : a) 0 to 2 sec, b) 2 s to 4 sec, c) 0 to 8 sec. POSITION-TIME GRAPH

19 Ans. : θ avg = x t = x f x i t f t i a) b) c)

20 Example Find the displacement, average velocity, and average speed of the car in the figure between positions A and F.

21 Ans. : Displacement: Average Velocity:

22 Ans. : Total distance traveled: From the car s position described by the curve, we find that the distance traveled from A to B is 22m plus 105m the distance traveled from B to F for a total of 127m.

23 Ans. : Applying the Formula to find Total Distance:

24 The instantaneous velocity The instantaneous velocity θ inst is the velocity right now, at this particular moment. If the velocity is constant : θ inst = θ avg SI unit : meter per second (m/s)

25 Acceleration We are often interested in how fast the velocity is changing. This is the acceleration. Acceleration is a change of velocity divided by a change of time. a avg = θ t = θ f θ i t f t i This quantity has units of (meters/second)/second. We will write this as m/sec 2 (there are no "square seconds"). As with the velocity, we can describe the instantaneous acceleration, the acceleration right now, at this particular moment. If the acceleration is constant : a inst = a avg

26 Acceleration If there is no change in speed, but there is a change in direction the acceleration has occured

27 Acceleration Example: Find the acceleration from the figure below f Ans. : a avg = θ t = 3 sec = 9 m/sec 3sec = 3 m/sec sec a avg = 3 m /sec = 3 m/sec2 sec

28 Velocity vs Acceleration When the object s velocity and acceleration are in the same direction, the object is speeding up (other hind the value for acceleration is positive). On the other hand, when the object s velocity and acceleration are in opposite direction, the object is slowing down. (also called deceleration)

29 MOTION DIAGRAMS ***

30 The Four Kinematic Equations

31 The Four Kinematic Equations θ = θ o + at x f = x i (θ + θ o)t a = θ θ o t θ θ o = at θ = θ o + at θ = x f x i t x f x i = θt θ = θ + θ o 2 x f x i = (θ + θ o) t 2 x f = x i (θ + θ o)t x f = x i + θ o t at2 x f = x i + (θ + θ o) t 2 θ = θ o + at x f = x i + (θ o + at + θ o) t 2 x f = x i + (2θ o + at ) t x 2 i + θ o t at2

32 The Four Kinematic Equations θ f 2 = θ o 2 + 2a (x f x i ) x f = x i + (θ + θ o) 2 x f = x i + (θ + θ o) t 2 a = θ θ o t = θ θ o t (θ θ o ) a a = x i + 1 2a θ2 θ2 o x f x i = 1 2a θ2 θ o 2 2a x f x i = θ 2 θ o 2 θ f 2 = θ o 2 + 2a (x f x i )

Ans. : θ xf = θ xi + a x t x f = x i + θ xi t + 1 2 a xt 2 θ xf = 0 + a x t θ

33 Example: Consider a car that starts at rest and accelerates at 2 m/s 2 for 3 seconds. At that time, t = 3 s, how fast is it going? and how far has it gone? Ans. : θ xf = θ xi + a x t x f = x i + θ xi t a xt 2 θ xf = 0 + a x t θ xf = a x t = 2 m/sec θ xf = 6 m/sec 3sec x = 1 2 at2 x = m/sec2 (3sec) 2 x = 9 m

34 Homework Problem 1: The velocity of a particle moving along the x-axis varies according to the expression v x = (40 5t 2 ) m/s, where t is in seconds. Find the average acceleration in the time interval t = 0 to t = 2.0 sec.

35 Homework Problem 2: Carrier Landing A jet lands on an aircraft carrier at 140 mi /h ( 63 m/s). a) What is its acceleration (assumed constant) if it stops in 2.0 sec? b) If the plane touches down at position x i = 0, what is the final position of the plane?

Distance vs. Displacement, Speed vs. Velocity, Acceleration, Free-fall, Average vs. Instantaneous quantities, Motion diagrams, Motion graphs,

Distance vs. Displacement, Speed vs. Velocity, Acceleration, Free-fall, Average vs. Instantaneous quantities, Motion diagrams, Motion graphs, Kinematic formulas. A Distance Tells how far an object is from