Analysis of Motion & Newton’s Laws

Class Objective zLearn and apply Newton’s First, Second, and Third Laws yUnidirectional yMultidirectional zLearn the relationship between position, velocity, and acceleration

RAT 10.1

Some Definitions (1D) Position - location on a straight line Displacement - change in location on a straight line x x  x = x 2 - x 1 = 4 - (-2) = 6

Some Definitions (1D) Average Velocity - rate of position change with time Instantaneous Velocity

Average and Instantaneous Velocity (1D) Position Time t 1 t 2 x1x1 x2x2 Slope = = Average Velocity x 2 - x 1 t 2 - t 1

Some Definitions (2D) Position -- a location usually described by a graphic on a map or by a coordinate system (-2, -3) (4, 3)

Some Definitions (2D) Displacement -- change in position, where (-2, -3) (4, 3) 5

Some Definitions (2D) Average velocity Instantaneous velocity Speed - the magnitude of instantaneous velocity (scalar)

Some Definitions (1D) Average Acceleration - rate of velocity change with time Instantaneous Acceleration

Average and Instantaneous Acceleration (1D) Velocity Time t 1 t 2 v1v1 v2v2 Slope = = Average Acceleration v 2 - v 1 t 2 - t 1

Some Definitions (2D) Average Acceleration Instantaneous Acceleration

Example One-dimensional motion Speed, miles per hour Time, hours

Paired Exercise What is the distance traveled? What is the acceleration at 1.25 hours? Speed, miles per hour Time, hours

For constant acceleration... if acceleration is constant integrating both sides v 0 is the original value at the beginning of the time interval (Definition)

Constant Acceleration substituting the velocity equation from the previous page integrating both sides yields (Definition)

Equations of Motion (Constant Acceleration) Velocity Position (in terms of x)

Multiple Directions Equations of motion can be written for each direction independently. Velocity Position

Distance, Velocity, and Acceleration zSuppose a dragster has constant acceleration. zIf a dragster starts from rest and accelerates to 60 mph in 10 seconds. How far did it travel?

Plot Speed vs time 60 mph 10 seconds time speed What does the area under the line represent? (1/6 min) (1 mi/min)

Distances…. Area = distance? Sure: Right? So:

Your Turn: RAT 10.2

Momentum v m p = m v momentum

Newton’s 1st Law: “Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.” “In the absence of a net force applied to an object, momentum stays constant.” Newton Holtzapple

Newton’s Second Law The time-rate-of-change of momentum is proportional to the net force on the object. If mass is constant...

Newton’s Third Law “To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal and directed in contrary parts.” Newton

Newton’s Third Law Other statements zForces always exist by the interaction of two (or more) bodies zThe force on one body is equal and opposite to the force on another body zIt is impossible to have a single isolated force acting in one direction zThe designation of an “action force” and a “reaction force” is arbitrary because there is mutual interaction between the bodies

Newton’s Third Law A Consequence The earth and the moon orbit about a common point about 1000 miles below the surface of the earth because the earth pulls on the moon and the moon pulls on the earth.

Example: Newton’s 3 rd Law zConsider a rocket with constant exhaust gas velocity: zThe mass changes (obviously) as the fuel is burned and the gas is ejected. fuel veve m v Positive

Example: Newton’s 3rd Law zThe magnitude of the net force acting on the rocket can be determined by observing its acceleration where m is the instantaneous mass of the rocket and dv is the instantaneous change in rocket velocity.

Example: Newton’s 3rd Law zThe magnitude of the net force acting on the ejected gas is where v e is the velocity of ejected gas and dm/dt is the rate mass is ejected from the rocket. (Note: The origin of this equation will become more clear when we do Accounting for Momentum.)

Example: Newton’s 3rd Law From Newton’s 3rd Law, these two forces must be opposite and equal to each other, so: or,

Example: 3rd Law Using calculus, this can be solved to yield: where m 0 is the initial mass of rocket including fuel

Why Newton’s Laws? zEngineers use models to predict things such as motion, fluid flow, lift on an airplane wing, movement of neutrons in a nuclear reactor, deflection of beams or columns, etc. zNewton’s laws are widely used and a good first example of engineering models.

More on Models Question: If I toss a piece of chalk at a sleeping student, does its path follow a parabola? Answer: Not exactly, because air resistance affects the motion. Also, we should consider the effect of the spinning earth as it moves around the sun in an ellipse. However, for most practical work, a parabola is close enough.