what is an instant of time? car on racetrack example:

What is an Instant of Time?What is an Instant of Time?

Car on racetrack example:Car on racetrack example:

1. How long did a car spend at any one 1. How long did a car spend at any one location?location?

2. Each position is linked to a time, but 2. Each position is linked to a time, but how long did that time last?how long did that time last?

3. You could say an “instant”, but how long 3. You could say an “instant”, but how long is that?is that?

CHAPTER – 5CHAPTER – 5A Mathematical Model of MotionA Mathematical Model of Motion5.1 Graphing Motion in One Dimension5.1 Graphing Motion in One Dimension Position – Time GraphsPosition – Time Graphs

Example:Example:Football running back motion (displacement) Football running back motion (displacement)

diagram at 1 second intervals.diagram at 1 second intervals.

Plot the Position/Time GraphPlot the Position/Time Graph

4. If an instant lasts for a finite amount of 4. If an instant lasts for a finite amount of time, then, because the car would be at time, then, because the car would be at the same position during that time, the car the same position during that time, the car would be at rest. would be at rest.

But, a moving object (car) cannot be at But, a moving object (car) cannot be at rest; rest;

an instant is not a finite period of time.an instant is not a finite period of time.

5. This means that an instant of time lasts 5. This means that an instant of time lasts “0” seconds.“0” seconds.

Using a Graph to Find Out Where Using a Graph to Find Out Where and When (Pick various locations)and When (Pick various locations)

Graphing the Motion of Two or Graphing the Motion of Two or More ObjectsMore Objects

A = running back A = running back C = centerC = center

B = linebackerB = linebacker D = defensive backD = defensive back

From Graphs to Words and Back From Graphs to Words and Back AgainAgain

Keep in mind that when t=0, the position of Keep in mind that when t=0, the position of the object does not necessarily have to be the object does not necessarily have to be zero.zero.

Uniform MotionUniform Motion

Definition of uniform motion =Definition of uniform motion =

Means that equal displacements occur during Means that equal displacements occur during successive equal time intervals.successive equal time intervals.

What does the “slope” of the pos/time What does the “slope” of the pos/time graph give us?graph give us?

Velocity Velocity

riserise ΔΔyy YYff - Y - Yii

slope = run = slope = run = ΔΔx = xx = xff - x - xii

ΔΔdd ddff - d - dii

slope = v = slope = v = ΔΔt = tt = tff - t - tii

For Objects With Diff. VelocitiesFor Objects With Diff. Velocities

Using an Equation to Find out Using an Equation to Find out Where and WhenWhere and When

ΔΔdd ddf f - d- dii

Average Velocity = v = Average Velocity = v = ΔΔt = tt = tff – t – tii

d = dd = dii + vt + vt

Example Problem (PP-11)Example Problem (PP-11)

A car starts 200m west of the town square A car starts 200m west of the town square and moves with a constant velocity of and moves with a constant velocity of 15m/s towards the east. 15m/s towards the east.

a) Where will the car be 10 min later?a) Where will the car be 10 min later?

b) When will the car reach the town b) When will the car reach the town square?square?

a) draw sketcha) draw sketch


d = -200m + (15ms)(600s)d = -200m + (15ms)(600s)

d = -200m + 9000md = -200m + 9000m

d = 8800md = 8800m

b)b) d = dd = dii + vt + vt

0 = -200m + (15m/s)t0 = -200m + (15m/s)t

200m = (15m/s)t200m = (15m/s)t

13.3s = t13.3s = t

Example-2 (PP 12)Example-2 (PP 12)

A car starts 200m west of the town square A car starts 200m west of the town square and moves with a constant velocity of and moves with a constant velocity of 15m/s towards the east. At the same time 15m/s towards the east. At the same time a truck was 400m east of the town square a truck was 400m east of the town square moving west at a constant velocity of moving west at a constant velocity of 12m/s. 12m/s.

Find the time and place where the car meets Find the time and place where the car meets the truck.the truck.

Draw a sketch.Draw a sketch.


5.2 Graphing Velocity in One 5.2 Graphing Velocity in One Dimension, Determining Dimension, Determining Instantaneous VelocityInstantaneous Velocity

Q: When an object is not moving with Q: When an object is not moving with uniform motion, the object is said to be…?uniform motion, the object is said to be…?

A: acceleratingA: accelerating

Position-Time Graph Uniform MotionPosition-Time Graph Uniform Motion

Position-Time Graph AccelerationPosition-Time Graph Acceleration

Instantaneous Velocity = ?Instantaneous Velocity = ?

How fast an object is moving at a How fast an object is moving at a particular instant in time, ie: how fast is it particular instant in time, ie: how fast is it moving “Right Now”moving “Right Now”

What is the instantaneous velocity at 2s?What is the instantaneous velocity at 2s?

What is the instantaneous velocity at 4s?What is the instantaneous velocity at 4s?

Instantaneous VelocityInstantaneous Velocity

The Instantaneous Velocity is equal to the The Instantaneous Velocity is equal to the “Slope” of the tangent line of a “Slope” of the tangent line of a position/time graph at any particular time.position/time graph at any particular time.

Draw previous graph and calculate the Draw previous graph and calculate the instantaneous velocity at 2s & 4s.instantaneous velocity at 2s & 4s.

Velocity-Time GraphsVelocity-Time Graphs

2 planes Plane-A & Plane-B2 planes Plane-A & Plane-B

vvBB is a constant 75m/s is a constant 75m/s

vvAA is constantly increasing (constant “a”) is constantly increasing (constant “a”)

Draw sketch on boardDraw sketch on boardQ: At the point of intersection, will the Q: At the point of intersection, will the planes crash?planes crash?A: ??? Not enough information given, the A: ??? Not enough information given, the graph merely indicates the planes have graph merely indicates the planes have the same velocity at that point.the same velocity at that point.

Displacement from a Velocity-Time GraphDisplacement from a Velocity-Time Graph

Q: What does the area under a V-T graph Q: What does the area under a V-T graph represent?represent?

A: A: ΔΔd, displacement.d, displacement.

5.3 Acceleration5.3 AccelerationDetermining Average AccelerationDetermining Average Acceleration

Average acceleration, “a” , is equal to the slope (rise/run, Average acceleration, “a” , is equal to the slope (rise/run, ΔΔv/v/ΔΔtt) of a velocity-time graph.) of a velocity-time graph.

Constant and Instantaneous Constant and Instantaneous AccelerationAcceleration

If there is a constant slope on a Velocity-Time If there is a constant slope on a Velocity-Time Graph then there is also a constant Graph then there is also a constant acceleration, any point “a” is the same. acceleration, any point “a” is the same.

Acceleration is simply the slope of the line.Acceleration is simply the slope of the line.

Instantaneous Acceleration of a Instantaneous Acceleration of a Velocity-Time Graph, Curved LineVelocity-Time Graph, Curved Line

Draw graph on board.Draw graph on board.

Q: What is the instantaneous acceleration Q: What is the instantaneous acceleration at 2s? How could it be determined?at 2s? How could it be determined?

A: Draw a tangent line at 2s then calculate A: Draw a tangent line at 2s then calculate the slope.the slope.

Positive and Negative AccelerationPositive and Negative Acceleration

VV11 speed increase/decrease/constant speed increase/decrease/constant

VV22 velocity +/- velocity +/-

VV33 acceleration +/-/0 acceleration +/-/0

VV44

VV55

VV66

When v+ and a+, speed increases(+)When v+ and a+, speed increases(+)

When v+ and a-, speed decreases(+)When v+ and a-, speed decreases(+)

When v- and a-, speed increases(-)When v- and a-, speed increases(-)

When v- and a+, speed decreases(-)When v- and a+, speed decreases(-)

Calculating Velocity from Calculating Velocity from AccelerationAcceleration

v = vv = voo + at + at

Example ProblemExample Problem

Hines Ward is running for a touchdown at Hines Ward is running for a touchdown at 4m/s. He accelerates for 3s. His velocity 4m/s. He accelerates for 3s. His velocity entering the end zone is 7m/s. What is his entering the end zone is 7m/s. What is his acceleration?acceleration?

v = vv = voo + at + at

7m/s = 4m/s +a(3s)7m/s = 4m/s +a(3s)

3m/s = a(3s)3m/s = a(3s)

1m/s1m/s22 = a = a

Displacement Under Constant Displacement Under Constant AccelerationAcceleration

d = dd = dii + ½(v + ½(vff + v + vii)t)t

d = dd = di i + v+ viit + 1/2att + 1/2at22

vv22 = v = vii22 + 2a(d + 2a(dff – d – dii))

5.4 Free Fall5.4 Free FallAcceleration Due to GravityAcceleration Due to Gravity

1. Drop a flat sheet of paper1. Drop a flat sheet of paper2. Drop a crumpled piece of paper.2. Drop a crumpled piece of paper.3. Drop a tennis ball.3. Drop a tennis ball.Q: Is there a difference in the acceleration Q: Is there a difference in the acceleration of each of the objects above?of each of the objects above?A: NO All “free falling” objects accelerate A: NO All “free falling” objects accelerate with a magnitude of 9.8m/swith a magnitude of 9.8m/s22 towards the towards the center of the Earth.center of the Earth.Acceleration due to gravity “g” = -9.8m/sAcceleration due to gravity “g” = -9.8m/s22

Example: Drop A RockExample: Drop A Rock

After 1s it is falling at ____m/sAfter 1s it is falling at ____m/s




Each second during free fall the rock will Each second during free fall the rock will ΔΔ its velocity by -9.8m/s. its velocity by -9.8m/s.

Drop a Rock DiagramDrop a Rock Diagram

DIAGRAMDIAGRAMDuring each equal successive time interval the During each equal successive time interval the

rock will fall a greater distancerock will fall a greater distance b/c a = “-g”b/c a = “-g”Q: Could this diagram also apply to a rock Q: Could this diagram also apply to a rock thrown upward?thrown upward?A: YES, the diagram would look the same.A: YES, the diagram would look the same.Q: Why?Q: Why?A: Once the rock leaves the hand, the only force A: Once the rock leaves the hand, the only force acting on the rock is gravity “g”.acting on the rock is gravity “g”.

Diagram/Example

Diagram of a ball thrown upward with a velocity of 49m/s.

Show the velocity changes for 10 seconds, 5 seconds up & 5 seconds down.

DRAW SKETCH

Time Velocity0s 49m/s1s 39.2m/s2s 29.4m/s3s 19.6m/s4s 9.8m/s5s 0m/s6s -9.8m/s7s -19.6m/s8s -29.4m/s9s -39.2m/s10s -49m/s

Example ProblemExample Problem

If you throw a rock upward with an initial If you throw a rock upward with an initial velocity of 35m/s:velocity of 35m/s:

a) what is its velocity after 1,2,3,4,5 sec?a) what is its velocity after 1,2,3,4,5 sec?

b) what is its position after 1,2,3,4,5 sec?b) what is its position after 1,2,3,4,5 sec?

c) how long will it take to reach its c) how long will it take to reach its maximum height?maximum height?

d) what is its maximum height?d) what is its maximum height?

e) how long will it be in the air? e) how long will it be in the air?

Example Thrown UpwardExample Thrown Upward

Quadratic EquationQuadratic Equation

y = axy = ax22 + bx + c + bx + c

When y = 0When y = 0

0 = ax0 = ax22 + bx + c + bx + c

Solving for “x”Solving for “x”

- b ± √(b- b ± √(b22 – 4ac) – 4ac)

x = 2ax = 2a

A ball is thrown upward with an initial A ball is thrown upward with an initial velocity of 35m/s, how long will the ball be velocity of 35m/s, how long will the ball be in the air?in the air?

Equation to be used…?Equation to be used…?

d = dd = dii + v + viit + 1/2att + 1/2at22

d = ½(a)td = ½(a)t22 + v + viit + dt + dii

0 = ½(-9.8m/s0 = ½(-9.8m/s22)t)t22 + (35m/s)t + 0 + (35m/s)t + 0

0 = (-4.9m/s0 = (-4.9m/s22)t)t22 + (35m/s)t + (35m/s)t

Solve for “t”Solve for “t”

- b + √(b- b + √(b22 – 4ac) – 4ac)

x = 2ax = 2a

-35m/s + √-35m/s + √﴾﴾(35m/s)(35m/s)22 – 4(-4.9m/s)(0) – 4(-4.9m/s)(0) ﴿ ﴿t = 2(-4.9m/st = 2(-4.9m/s22))

-35m/s + √1225m-35m/s + √1225m22/s/s22

t = -9.8m/st = -9.8m/s22

-35m/s + 35m/s-35m/s + 35m/s 0__ 0__

t = -9.8m/st = -9.8m/s22 = -9.8m/s = -9.8m/s22 = 0s = 0s

OR →OR →

- b - √(b- b - √(b22 – 4ac) – 4ac)

x = 2a x = 2a -35m/s - √-35m/s - √﴾﴾(35m/s)(35m/s)22 – 4(- – 4(-4.9m/s)(0)4.9m/s)(0) ﴿ ﴿t = 2(-4.9m/st = 2(-4.9m/s22))

-35m/s - √1225m-35m/s - √1225m22/s/s22

t = -9.8m/st = -9.8m/s22

-35m/s - 35m/s-35m/s - 35m/s -70m/s-70m/s

t = -9.8m/st = -9.8m/s22 = -9.8m/s = -9.8m/s22

t = 7.143s t = 7.143s

what is an instant of time? car on racetrack example:

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