Chapter 5 A Mathematical Model of Motion Physics Chapter 5 A Mathematical Model of Motion

A Mathematical Model of Motion 5.1 Graphing Motion in One Dimension 5.2 Graphing Velocity in One Dimension 5.3 Acceleration 5.4 Free Fall

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

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

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

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

Position-Time Graphs Slope of a position-time graph gives you the average velocity of the object Slope = rise/run = y/x = (d1- do)/(t1- to) Slope = meters/seconds = average velocity

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

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

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

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

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

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

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

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

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

Position-Time Graphs 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) = (100-20) (11-5) = 13.3 m/s at 6 s

v = d/t = (d1 – do)/(t1 – to) Velocity Equations Uniform motion can be represented by algebraic equations Average Velocity v = d/t = (d1 – do)/(t1 – to)

Position Equations v = (d1 – do)/t1 d1 = do + vt Since most motion starts at to = 0 v = (d1 – do)/t1 or d1 = do + vt

Graphing Motion in One Dimension Summary Uniform motion Equal displacements during equal time intervals 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) Position equation d1 = do + vt

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

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

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

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

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

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

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

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

Graphing Velocity in One Dimension Slope = rise/run = v/t = (m/s)/s = m/s2 a = (v1 – vo)/(t1 – to)

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

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

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

Graphing Velocity in One Dimension Summary 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 Acceleration—the rate of change of velocity An acceleration-Time graph gives an object’s acceleration rate at any time

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

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

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

Acceleration The area under the curve for an acceleration-time graph gives you an object’s change in velocity Area = b x h = (m/s2)(s) = m/s

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

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

Acceleration Summary 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 acceleration Displacement Acceleration-Time (A-T) Acceleration at any time Junk Change in velocity

Equations of Motion for Uniform Acceleration d1 = do + v t v1 = vo + āt d1 = do + vot + 1/2āt2 v1 2 = v02 + 2ā(d1 -do) d1 = do + ½(v1 + vo)t Also…. v = 1/2(v1 + vo) v = d/t ā = v/t

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

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

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

Free Fall Actually without air resistance (in a vacuum) all objects fall at the same rate due to gravity Acceleration due to gravity (g) = - 9.8 m/s2 On Earth Downward (-)

Free Fall Acceleration due to gravity (g) = - 9.8 m/s2 This means that as any object falls its velocity increases by - 9.8 m/s each second. An object that moves through the air only under the influence of gravity is said to be in free fall.

Free Fall If an object is dropped, each second its downward velocity increases by 9.8 m/s After 1 sec; - 9.8 m/s After 2 sec; - 19.6 m/s After 3 sec; -29.4 m/s After 4 sec ????

Free Fall If an object is thrown upward, each second its downward velocity increases by 9.8 m/s If it starts with 19.6 m/s of upward velocity After 1 sec; loses 9.8 m/s, so is going 9.8 m/s After 2 sec; loses another 9.8 m/s, so stops (0 m/s) After 3 sec; gains -9.8m/s, so is going –9.8 m/s After 4 sec ????