Describing motion: Kinematics in one dimension

Transcription

1 Describing motion: Kinematics in one dimension Scientist Galileo Galilei Issac Newton Vocabulary Mechanics Kinematics Dynamics Translational Motion Particle Frame of Reference Coordinate axes Position Displacement Vectors Average speed Speed Velocity Average Velocity Instantaneous velocity Acceleration Average Acceleration Instantaneous Acceleration Slope Part 1 Kinematics deals with the description on how objects move Dynamics deals with the forces and why objects move as they do Translational motion- move without rotating We will look first at motion in one dimension A particle can undergo only translational motion 2-1 Reference Frames and Displacement Anytime you take a measurement you must have a frame of reference For instance a person on a train is walking towards the front of the train at 5km/h. If the train is going 80 km/hr then in respect to the ground the person has a speed of 85 km/h In everyday life our frame of reference is the earth When specifying the motion of an object, it is important to specify not only speed but also the direction of the motion cardinal directions- north, south, east, west up and down In physics- coordinate axes (x, y) To the right of the zero is considered to be positive (also up) To the left of the zero is considered to be negative (also down) 1 RoessBoss :29:54 1/8 Kinematics 1D.pdf (#2)

2 There is also a third dimension- z- we will not be using it in this class In this unit we will be focused on the x axes- 1 dimension Distance versus Displacement displacement is a change in position- how far away from the starting point Distance is the total amount and is independent of the starting point Displacement is a quantity that has both magnitude and direction Quantities with magnitude and direction are called vectors Vectors- are represented by arrows in diagrams- direction shows if it is positive or negative in value The symbol means change and is called delta Change is always determined by doing the final minus initial 2-2 Average Velocity Speed- refers to how far an object travels in a given time interval regardless of direction Average speed- the total distance traveled along its path divided by the time it takes to travel this distance Speed is simply a positive number with units Velocity- signifies both direction and magnitude (numerical value) Velocity is a vector Average velocity is defined in the terms of displacement and not distance + or can signify the direction for linear motion Equation Box Instantaneous Velocity Instantaneous velocity- is the velocity at any moment or the average velocity over an infinitesimally short time interval Therefore delta t is approaching zero If delta t was zero then the value would be undefined Note that the instantaneous speed always equals the magnitude of the instantaneous velocity- due to the distance and displacement becoming the same when they become infinitesimally small If the object moves at a uniform (constant) velocity over a particular interval then its instantaneous velocity at any instance is the same as its average velocity. The slope is a ratio of the displacement/time A slope chord joining 2 points on x vs t graph equals average velocity 2 RoessBoss :29:54 2/8 Kinematics 1D.pdf (2/8)

3 The definition of the instantaneous velocity is the limiting value of the average velocity as delta t approaches zero Thus the instantaneous velocity equals the slope of the tangent to the curve at that point In calculus we write it as the derivative (dx/dt) If an object moves with constant velocity over a particular time interval, its instantaneous velocity is equal to its average velocity In this case the graph will be a straight line and the slope equals the velocity (x vs t) Equation Box Acceleration An object whose velocity is changing is said to be accelerating Acceleration is how rapidly the velocity of an object is changing Average Acceleration is defined as the change in velocity divided by the time taken to make this change Acceleration is also a vector For one direction we only need to use a plus or minus to indicate direction relative to a chosen coordinate system Acceleration tells us how fast the velocity changes Velocity tells us how fast the position changes Deceleration does not necessarily mean that the acceleration is negative You can be moving in the positive direction but decreasing in velocity In deceleration your velocity and acceleration are opposite of each other Equation Box 2-3 Instantaneous Acceleration a is the limiting value of the average acceleration as we let delta t approach zero The limit dv/dt is the derivative of the v with respect to t 3 RoessBoss :29:54 3/8 Kinematics 1D.pdf (3/8)

4 In a graph of velocity vs time- the average acceleration over a time interval is represented by the slops of the straight line connecting two points in a position versus time graph- the slop represents average velocity The instantaneous acceleration at any time is the slope of the tangent to the v vs t curve at that time Like velocity- acceleration is a rate Velocity is a rate at which its displacement changes with time Its acceleration on the other hand is a rate at which its velocity changes with time Acceleration is a rate of a rate 4 RoessBoss :29:54 4/8 Kinematics 1D.pdf (4/8)

5 Equation Box Motion at Constant Acceleration The acceleration due to gravity near the earth s is an example of a=constant Now look at the magnitude- direction is constantly changing so it is accelerating If the object was in a straight line and acceleration was constant then it is not accelerating since acceleration means a change in velocity over time Rewrite the acceleration equation by moving isolating v Equation Box 2-5 Now look at the Velocity equation and isolate x Equation Box 2-6 Because the velocity increases at a uniform rate, the average velocity will be midway between the initial and the final velocities So average velocity (when acceleration is constant) Equation Box RoessBoss :29:55 5/8 Kinematics 1D.pdf (5/8)

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7 So know lets look at how we can take the equation from 2-6 as the base and sub in 2-7. Simplify it Equation Box 2-8 Now rearrange 2-5 for t Equation Box 2-9 Then take 2-8 and 2-9 and simplify Equation Box 2-10 All of these equations are used for constant acceleration They are not valid if a is not constant Many cases we can set the initial position as zero Remember that the difference in x is displacement and not distance 7 RoessBoss :29:55 7/8 Kinematics 1D.pdf (7/8)

8 2-7 Falling Objects A free falling object is undergoing acceleration Galileo postulated that all objects fall with the same constant acceleration in the absence of air or other resistance He showed that for an object falling from rest the distance traveled will be proportional to the square of the time Galileo also said that sir acts as a resistance to very light objects that have a large surface area Most ordinary circumstances the air resistance is negligible In chambers with air vacuumed out the objects fall at the same rate Basically- at a given location on Earth and in the absence of air resistance, all objects fall with the same constant acceleration We call this acceleration- the acceleration due to gravity g=9.80 m/s2 Acceleration due to gravity is a vector- as in any acceleration Acceleration due to gravity is downward toward the center of the Earth It is arbitrary whether we choose y to be positive in the upward direction or in the downward direction; but we must be consistent about it throughout a problem s solution 8 RoessBoss :29:55 8/8 Kinematics 1D.pdf (8/8)