Chapter 2 Motion Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

What is Motion?

Describing Motion Three basic concepts Applications Position Speed and velocity Acceleration Applications Horizontal motion (along a line) Vertical motion (falling objects) Compound motion (2-D)

Explaining Motion Basic ideas Applications Forces Momentum and impulse Inertia and mass Newton’s laws Applications Momentum and impulse Circular motion Newton’s law of gravitation Earth satellites

Measuring Motion Two fundamental components: Change in position Change in time Three important combinations of length and time: Speed Velocity Acceleration

Speed Change in position with respect to time Average speed - most common measurement Instantaneous speed - time interval approaches zero

Example: Average Speed Calculate average speed between trip times of 1 h and 3 h

Velocity Describes speed (How fast is it going?) and direction (Where is it going?) Graphical representation using vectors: length = magnitude; arrowheads = direction

Acceleration Rate at which motion changes over time Speed can change Direction can change Both speed and direction can change

Movement relative to some reference, usually a stationary object. Problem (A) 11 How long will be required for a car to go from a speed of 20.0 m/s to a speed of 25.0 m/s if the acceleration is 3.0 m/s2? What is Motion? Movement relative to some reference, usually a stationary object.

Uniform Acceleration Constant, straight line acceleration Average velocity simply related to initial and final velocities in this case

Movement relative to some reference, usually a stationary object. Example Problem 2.6 Find acceleration (deceleration) of the automobile in Example 2.5 (car moving at 25.0 m/s and stops in 10.0 seconds). What is Motion? Movement relative to some reference, usually a stationary object.

Forces - Historical Background Aristotle Heavier objects fall faster Objects moving horizontally require continuously applied force Relied on thinking alone Galileo and Newton All objects fall at the same rate No force required for uniform horizontal motion Reasoning based upon measurements

Force A push or pull capable of changing an object’s state of motion Overall effect determined by the (vector) sum of all forces - the “net force” on the object

Fundamental Forces Most basic of all interactions Gravitational Mass interactions Motions of planets, stars, galaxies… Electromagnetic Charge interactions Electricity and magnetism Atoms and molecules, chemistry 3. Weak force Involved in certain nuclear reactions 4. Strong force Holds nuclei together

Horizontal Motion on Land “Natural motion” question: Is a continuous force needed to keep an object moving? No, if there are no unbalanced retarding forces Inertia - measure of an object’s tendency to resist changes in its motion (including rest)

Balanced and Unbalanced Forces Motion continues unchanged w/o unbalanced forces Retarding force decreases speed Boost increases speed Sideways force changes direction

Falling Objects Free fall - falling under influence of gravity w/o air resistance Distance proportional to time squared Speed increases linearly with time Trajectories exhibit up/down symmetries Acceleration same for all objects

Movement relative to some reference, usually a stationary object. Problem (A) 16 A ball thrown straight up climbs for 3.0 s before falling. Neglecting air resistance, with what velocity was the ball thrown? What is Motion? Movement relative to some reference, usually a stationary object.

Movement relative to some reference, usually a stationary object. Free Fall Acceleration due to gravity (free fall neglecting air resistance) g = 9.8 m/s2 (downward) = 32 ft/s2 (downward) What is Motion? Movement relative to some reference, usually a stationary object.

Summary of Motion Equations (see page 55) Average velocity vavg = d / t vavg = (vf + vi) / 2 Acceleration a = (vf - vi) / t vf = vi + at t = (vf - vi) / a Free fall distance from rest d = ½at2 where, a = g = 9.8 m/s2 What is Motion? Movement relative to some reference, usually a stationary object.

Compound Motion Three types of motion: Vertical motion Horizontal motion Combination of 1. and 2. Projectile motion An object thrown into the air Basic observations: Gravity acts at all times Acceleration (g) is independent of the object’s motion

Projectile Motion Vertical projectile Slows going up Stops at top Accelerates downward Force of gravity acts downward throughout Horizontal projectiles Horizontal velocity remains the same (neglecting air resistance) Taken with vertical motion = curved path

Fired Horizontally Versus Dropped Vertical motions occur in parallel Arrow has an additional horizontal motion component They strike the ground at the same time!

Example: Passing a Football Assume only force = gravity (down) Vertical velocity decreases, stops and then increases Horizontal motion is uniform Combination of two motions = parabola

Three Laws of Motion First detailed by Newton (1564-1642 AD) Concurrently developed calculus and a law of gravitation Essential idea – forces

Newton’s 1st Law of Motion “Law of inertia” Every object retains its state of rest or its state of uniform straight-line motion unless acted upon by an unbalanced force Inertia resists any changes in motion

Newton’s 2nd Law of Motion “F = ma” The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to the mass of the object. Depends both on the net force applied and the mass of the object.

Newton’s 2nd Law of Motion Forces cause accelerations Units = Newtons (N), kg-m/s2 Proportionality constant = mass More force, more acceleration More mass, less acceleration

Examples - Newton’s 2nd More mass, less acceleration She’s not pedaling where, m = mb + mg

Examples - Newton’s 2nd Focus on net force Net force zero here Air resistance + tire friction match applied force Result: no acceleration; constant velocity

Movement relative to some reference, usually a stationary object. Problem (A) 19 What is the resulting acceleration when an unbalanced force of 100 N is applied to a 5 kg object? What is Motion? Movement relative to some reference, usually a stationary object.

Weight and Mass Mass = quantitative measure of inertia; the amount of matter Weight = force of gravity acting on the mass Pounds and newtons measure of force Kilogram = measure of mass

Newton’s 3rd Law of Motion Source of force - other objects 3rd law - relates forces between objects “Whenever two objects interact, the force exerted on one object is equal in size and opposite in direction to the force exerted on the other object.”

Momentum Important property closely related to Newton’s 2nd law Includes effects of both velocity (motion) and mass (inertia)

Conservation of momentum The total momentum of a group of interacting objects remains the same in the absence of external forces Applications: analyzing action/reaction interactions, collisions

Movement relative to some reference, usually a stationary object. Problem (A) 22 A 15 g bullet is fired with a velocity of 200 m/s from a 6 kg rifle. What is the recoil velocity of the rifle? What is Motion? Movement relative to some reference, usually a stationary object.

Impulse A force acting on an object for some time t An impulse produces a change in momentum Applications: padding for elbows and knees, airbags, protective plastic barrels on highways

Forces and Circular Motion Circular motion = accelerated motion (direction changing) Centripetal acceleration present Centripetal force must be acting Centrifugal force - apparent outward tug as direction changes If centripetal force ends: motion = straight line

Movement relative to some reference, usually a stationary object. Problem (A) 30 How much centripetal force is needed to keep a 0.20 kg ball on a 1.50 m string moving in a circular path with a speed of 3.0 m/s? What is Motion? Movement relative to some reference, usually a stationary object.

Newton’s Law of Gravitation “Universal Law of Gravitation” Every object in the universe is attracted to every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distances between them. F = G m1m2 / d2 (Eq. 2.12)

Newton’s Law of Gravitation Attractive force between all masses Directly proportional to product of the masses Inversely proportional to separation distance squared Explains why g=9.8m/s2 Provides centripetal force for orbital motion

Application of Forces Fdrag = f(area,v,ρatm) Fmass = mg Fnet = ΣFi = manet -Fthrust = Fupward = Mass decreases with time Fthrust

Application of Forces At liftoff: gross mass = 2.04 x 106 kg total thrust = 30.18 MN assume drag = 0 N Fmass = - mg = -(2.04x106)(9.81) Fupward = -Fthrust = 30.18 MN Fnet = ΣFi = manet = 10.17 MN anet = 5.0 m/s2 d(1s) = ½ at2 = 2.5 m (~8.2 ft) Mass change = 10,400 kg/s