Physics Chapter 5 A Mathematical Model of Motion

Physics Chapter 5 A Mathematical Model of Motion

A Mathematical Model of Motion Ü 5. 1 Graphing Motion in One Dimension Ü 5. 2 Graphing Velocity in One Dimension Ü 5. 3 Acceleration Ü 5. 4 Free Fall

Graphing Motion in One Dimension Ü Motion can be described in words, sketches, diagrams, graphs and equations Ü Graphs of motion can be read to determine various characteristics of motion Ü Motion graphs have time on the independent (x) axis and either position, velocity or acceleration on the dependent (y) axis Ü Reading these graphs can give you an object’s position, velocity and/or acceleration at any time during its motion

Position-Time Graphs Ü These graphs show an object’s position at any time (instant) Ü Position—where an object is located relative to a reference point Ü Y (position) versus X (time)

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü This graph tell you about the object’s position and it also tells you about its speed (velocity) Ü Velocity—the rate of change of an object’s position (displacement) for a time interval

Position-Time Graphs Ü Slope of a position-time graph gives you the average velocity of the object Ü Slope = rise/run = y/ x = (d 1 - do)/(t 1 - to) Ü Slope = meters/seconds = average velocity

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü How are the graphs below different? Ü What do they tell you about the object’s motion?

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü So far, all the graphs have been of uniform motion— equal displacements occur over equal successive time intervals Ü Let’s see when this is not the case

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü What does this graph tell you about the object’s motion?

Position-Time Graphs Ü For this graph we can find the instantaneous velocity of the object at any time Ü But without calculus, we can only find the slope of a straight line Ü To to this we must draw a straight line called a tangent line Ü Tangent line—a line that hits a curve at only one point and doesn’t cross the line Ü The slope of this tangent line will give use the velocity of the object at that instant

Position-Time Graphs Ü At 6 seconds we draw a tangent line and then find the slope of that line to find the velocity at that time only (instantaneous velocity)

Position-Time Graphs Slope = rise/run = y/ x = (100 -20) (11 -5) = 13. 3 m/s at 6 s

Velocity Equations Ü Uniform motion can be represented by algebraic equations Ü Average Velocity Üv = d/ t = (d 1 – do)/(t 1 – to)

Position Equations Ü Since most motion starts at to = 0 v = (d 1 – do)/t 1 or d 1 = do + vt

Graphing Motion in One Dimension Summary Ü Uniform motion Equal displacements during equal time intervals Ü Position-Time graph Reading it tells you an object’s position at any time Slope of line (curve) gives you the object’s velocity (average or instantaneous) Ü Position equation d 1 = do + vt

Graphing Velocity in One Dimension ÜA velocity-time graph gives you an object’s velocity at any time Ü This graph is my favorite graph — reading it gives you the most motion information of any graph!

Graphing Velocity in One Dimension Ü What does this graph tell you about an object’s motion?

Graphing Velocity in One Dimension Ü What does this graph tell you about an object’s motion?

Graphing Velocity in One Dimension Ü What does this graph tell you about an object’s motion?

Graphing Velocity in One Dimension Ü In which graph is the object moving faster?

Graphing Velocity in One Dimension Ü What does this graph tell you about an object’s motion?

Graphing Velocity in One Dimension Ü Is the velocity constant in this graph? Ü How do you know?

Graphing Velocity in One Dimension Ü The slope of a P-T graph gives you an object’s velocity Ü The slope of a Velocity-time graph gives you an object’s acceleration Ü Acceleration—the rate of change in an object’s velocity during a time interval

Graphing Velocity in One Dimension Ü Slope = rise/run = v/ t = (m/s)/s = m/s 2 Ü a = (v 1 – vo)/(t 1 – to)

Graphing Velocity in One Dimension Ü You can also find an object’s displacement from a Velocity-Time graph Ü Area = b x h (or ½ bh) Ü Area = (m/s)(s) Ü Area = meters

Graphing Velocity in One Dimension Ü What does this graph tell you about an object’s motion?

Graphing Velocity in One Dimension Ü Find the velocity at 4 seconds Ü Find the average acceleration and displaceme nt between 0 and 4 seconds

Graphing Velocity in One Dimension Summary Ü Velocity-Time Graph Reading the graph gives you an object’s velocity at any time Slope of V-T graph gives you an object’s acceleration Area under the curve of a V-T graph gives you an object’s displacement

Acceleration Ü Acceleration—the rate of change of velocity Ü An acceleration-Time graph gives an object’s acceleration rate at any time

Acceleration Ü What does this graph tell you about the object’s motion?

Acceleration Ü What does this graph tell you about the object’s motion?

Acceleration Ü What does this graph tell you about the object’s motion?

Acceleration Ü The area under the curve for an acceleration-time graph gives you an object’s change in velocity Ü Area = b x h = (m/s 2)(s) = m/s

Acceleration Ü Find the acceleration rate at 2 seconds. Ü Find the change in velocity of the object between 2 and 6 seconds

Acceleration Ü What does this graph tell you about the object’s motion?

Acceleration Summary Ü Acceleration-Time Graph Reading the graph gives you an object’s acceleration at any time Area under the curve of a A-T graph gives you an object’s change in velocity

Graph Summary Graph Read it Slope it Area it Position. Time (P-T) Position at any time Average velocity Garbage Velocity. Time (V-T) Velocity at any time Average Displacement acceleratio Acceleration Acceleratio n. Junk Change in -Time (A-T) n at any velocity time

Equations of Motion for Uniform Acceleration Ü d 1 = do + v t Ü v 1 = vo + āt Ü d 1 = do + vot + 1/2āt 2 Ü v 1 2 = v 02 + 2ā(d 1 -do) Ü d 1 = do + ½(v 1 + vo)t Also…. Ü v = 1/2(v 1 + vo) Ü v = d/ t Ü ā = v/ t

Equations of Motion for Uniform Acceleration Click here for Chapter 5 Problem Solving: New Equations

Free Fall Ü When you drop a feather and a rock from the same height, which one hits the ground first? Ü Why? Ü Book and paper demo

Free Fall Ü Do heavier objects fall faster than lighter objects? Ü How can I make a piece of paper fall faster/slower? Ü Which has more air resistance; an elephant or a snowflake? Ü Why doesn’t an elephant fall slower than a snowflake?

Free Fall Ü Actually without air resistance (in a vacuum) all objects fall at the same rate due to gravity Ü Acceleration due to gravity (g) = - 9. 8 m/s 2 Ü On Earth Ü Downward (-)

Free Fall Ü Acceleration due to gravity (g) = - 9. 8 m/s 2 Ü This means that as any object falls its velocity increases by - 9. 8 m/s each second. Ü An object that moves through the air only under the influence of gravity is said to be in free fall.

Free Fall Ü If an object is dropped, each second its downward velocity increases by 9. 8 m/s Ü After 1 sec; - 9. 8 m/s Ü After 2 sec; - 19. 6 m/s Ü After 3 sec; -29. 4 m/s Ü After 4 sec ? ?

Free Fall Ü If an object is thrown upward, each second its downward velocity increases by 9. 8 m/s Ü If it starts with 19. 6 m/s of upward velocity Ü After 1 sec; loses 9. 8 m/s, so is going 9. 8 m/s Ü After 2 sec; loses another 9. 8 m/s, so stops (0 m/s)