1D Motion Physics 2015-2016

Vector and Scalar Quantities

• Vector quantities need both magnitude (size) and direction to completely describe them– Ex: Displacement, Velocity, Acceleration

• Scalar quantities are completely described by magnitude only– Ex: Distance travelled, speed, Energy, time

Position

f ix x x

Frame of Reference Practice

• Describe your position in three different reference frames.

Displacement

Displacement Isn’t Distance

• The displacement of an object is not the same as the distance it travels– Ex: Throw a ball straight up and then catch it at

the same point you released it• The distance is twice the height – 2h• The displacement is zero

h

Position, Displacement and Distance travelled – Practice WB

State the position of the car at:

A: ____ E: ____

State the distance travelled from:

A to B: _____A to D: _____A to F: ______

State the displacement of the car from:

A to B: _____A to D: _____A to F: _____

Write:

• What is the difference between position, distance travelled and displacement.

Average Speed

• The average speed of an object is defined as the total distance traveled divided by the total time elapsed

• Scalar quantity• Ignores any variations in the object’s actual motion during the

trip• The total distance and the total time are all that is important• SI units = m/s

total distance

Average speedtotal time

Average Velocity

• The average velocity is rate at which the displacement occurs

• Direction will be the same as the direction of the

displacement (time interval is always positive)– + or - is sufficient for now

• SI Units of velocity are m/s– (may need to be converted)– Ignores actual movement as well

fi

averagefi

x xxv

t t t

Average Speed and Velocity - Practice - wb

State the Average Speed if it takes 20 seconds to travel :

From A to B: ______

From A to D: ______

From A to F: _______

From F to A: _______

State the Average Velocity if it takes 20 seconds to travel:

From A to B: _______

From A to D: _______

From A to F: ________

From F to A:

Speed vs. Velocity

Cars on both paths have the same average velocity since they had the same displacement in the same time interval

The car on the blue path will have a greater average speed since the distance it traveled is larger

Practice #1 - WB

An athlete swims from the north end to the south end of a 50.0 m pool in 20.0 seconds and makes the return trip to the starting position in 22.0 s.› What is the average velocity for the first half of the

swim? ________› What is the average velocity of the second half of

the swim? _______› What is the average velocity for the round trip?

_______

Practice #2

• Example: Two friends bicycle 3.0 kilometers north and then turn to bike 4.0 kilometers east in 25 minutes.

• a) What is their average speed?

• b) What is their average velocity?

Uniform Motion/Constant Velocity

• Uniform motion = constant velocity : a =0• The displacement changes by equal amounts

in equal intervals of time

Time (s) 0 1 2 3 4

Distance (m)

Velocity (m/s)

Acceleration (m/s2)

HP: Uniform Motion Formula

• Displacement at t = 0 is xo

Displacement at time t is x

And,

Then: x = xo+vt

xo x

t = 0 t = t

0ox xx

vt t

HP: Uniform Motion Examples

1. a. A car initially at position -5.0 m has a uniform velocity of 7.0 m/s. – At what position is it 4.0 seconds later?

b. A car initially at position 4 m has a velocity of -3 m/s. – At what time does it reach a position of -20

meters?– What distance does the car travel in this time?

Physics 500 Lab

• Regular Physics only

HP - Instantaneous Velocity

• The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero

• The instantaneous velocity indicates what the velocity is at every point of time, not just over a period of time

• Example: Car is speeding up and slowing down – average velocity doesn’t tell us much about its movement.

lim

0

x v=t

xv or

t t

Position vs. Time Graphs

• Computer/Vernier in groups

Graphical Interpretation:distance vs. time - wb

s/m

t/s

•Not Moving•Constant velocity•Constant velocity of 5ms-1 •Negative velocity of 5ms-1 •Starting from rest and accelerating•Starting 20 meters away at a fast pace and slowing down.

Graphical Interpretation of Velocity

• Velocity can be determined from a position-time graph

• Average velocity equals the slope of the line joining the initial and final positions of a distance – time graph. UNITS can prove it!

• An object moving with a constant velocity will have a graph that is a straight line

*Average Velocity, Constant

• The straight line indicates constant velocity

• The slope of the line is the value of the average velocity

Average Velocity, Non Constant

The motion is non-constant velocity

The average velocity is the slope of the blue line joining two points

HP - Instantaneous Velocity on a Graph

• The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time– The instantaneous speed

is defined as the magnitude of the instantaneous velocity

100m Split times - discussion

Acceleration

or fiv v v

a at t

Now Try Practice B

Velocity vs. Time graphs

• Computer/Vernier in groups

Graphical Interpretation: velocity vs. time

• Not moving• Constant velocity• Accelerating• Slowing down

v/ms-1

t/s

HP - Using Velocity vs. time graphs to calculate displacement.

•The area below a velocity – time graph represents the total displacement between those two times.• Prove it with UNITS!•Example:

HP - Graphical Interpretation of Acceleration

• Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph – UNITS can prove it!

• HP - Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

Average Acceleration

Uniformly Accelerated Motion

Time (s) 0 1 2 3

Distance (m) 0 3 10 23

Velocity (m/s) 0 5 10 15

Acceleration (m/s2) 5 5 5 5

•Uniformly Accelerated Motion = Constant Acceleration•The velocity changes equal amounts in equal amounts of time.

HP - Some practice

For each simulation, state whether each is positive, negative or zero.

Displacement Velocity Acceleration

HP - Some Practice

Quiz #1

• Definitions and finding distance, displacement, speed, velocity, acceleration

• Calculations of average velocity (Practice A) and average acceleration (Practice B)

• Interpreting, creating and finding values from position vs. time and velocity vs. time graphs

Problem-Solving Requirements• Read the problem• Draw a diagram!!

– Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations

• Label all known quantities, be sure all the units are consistent– Convert if necessary

• Choose the appropriate kinematic equation(s) that has only one variable that you don’t have the value for.

• Solve for the unknowns– You may have to solve two equations for two unknowns. Be careful of

Negatives!• Check your results

– Estimate and compare– Check units

Constant Acceleration Formula

0fiv vv

at t

2. A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8 m/s2 for 15 s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

Constant Acceleration Example

3. An object starting with an initial velocity of 5.0 m/s undergoes constant acceleration. After 10 seconds its velocity is found to be 35 m/s.– What is the acceleration?– Is it possible to know its position?

4. An object starting with an initial velocity of 2 m/s undergoes a constant acceleration of 4 m/s2. – When does it reach a velocity of 14 m/s?Now Try: Practice D

Another Constant Acceleration Formula

Example

5. A racing car reaches a speed of 42 m/s. It then begins a uniform negativeacceleration, using its parachute and braking system, and comes to rest5.5 s later. Find the distance that the car travels during braking.

Now Try: Practice C

Another Constant Acceleration Formula

6. A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500 m/s2. What is the velocity of the stroller after it has traveled 4.75 m?

•Now Try Practice E

Relationship Between Acceleration and Velocity

• Uniform velocity (shown by red arrows maintaining the same size)

• Acceleration equals zero

Relationship Between Velocity and Acceleration

Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same

length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration

Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same

length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

HP - Uniformly Accelerated Motion - Equations

• Used in situations with uniform/constant acceleration and uniform/constant acceleration ONLY!!!!!

2

2 2

Derived Formulas:

2122 ( )

i f

i

fi

v vx t

x v t at

v v a x

From Definitions:

2

ave

fi

fiave

s v t

v v at

v vv

Deriving the equations

HP - Examples:

• 7. Nathan accelerates his skateboard uniformly along a straight path from rest to 12.5 m/s in 2.5 sec.

A. What is Nathan’s displacement during this time interval?

B. What is Nathan’s average velocity during this time interval?

HP - Example:

8. A body at the origin has an initial velocity of -6.0 ms-1 and moves with an acceleration of 2.0 ms-2 . When will its displacement become 16 m?

HP - Example:

9. A car starts from rest and travels for 5 seconds with a uniform acceleration of +1.5m/s2. The driver then applies the brakes, causing a uniform acceleration of -2 m/s2. If the brakes are applied for 3 seconds, how fast is the car going at the end of the braking period, and how far has it gone from its start?

Frames of Reference

• Video• We are used to measuring velocities with

respect to observers who are “at rest” however, if you are moving, you are in your own frame of reference and observe things accordingly.

Relative Motion and Frames of Reference

• You are waiting for a bus • A bus passes you at 20 m/s• A passenger is walking on the bus at a speed

of 1 m/s – What is his relative velocity if he walks in the

direction the bus is going– What is his relative velocity if he walks in the

direction opposite the bus.

Relative Motion and Frames of Reference Example

1. Car 1 – 35 km/hr Car 2 – 40 km/hr Car 2 overtakes Car 1 because it is going faster What is the relative speed of Car 1 to Car 2

Free Fall and Air resistance

Galileo Galilei

• 1564 - 1642• Galileo formulated the

laws that govern the motion of objects in free fall

• Also looked at:– Inclined planes– Relative motion– Thermometers– Pendulum

Free Fall

• Video• Galileo’s thought experiment• All objects moving under the influence of gravity only

are said to be in free fall– Free fall does not depend on the object’s original motion

• All objects falling near the earth’s surface fall with a constant acceleration

• The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity

• Symbolized by g• g = 9.8 m/s² or 9.81 m/s2

– When estimating, use g 10 m/s2

• g is always directed downward– toward the center of the earth – typically negative in the

equations• Ignoring air resistance and assuming g doesn’t vary

with altitude over short vertical distances, free fall is constantly accelerated motion

Free Fall – an object dropped

• Initial velocity is zero• Let up be positive• Use the kinematic

equations– Generally use y instead

of x since vertical• Acceleration is

g = -9.80 m/s2

U = 0

a = g

Free Fall – an object thrown downward

• a = g = -9.80 m/s2

• Initial velocity 0– With upward being

positive, initial velocity will be negative

Free Fall -- object thrown upward

• Initial velocity is positive• The instantaneous

velocity at the maximum height is zero

• a = g = -9.80 m/s2 everywhere in the motion

v = 0

Thrown upward, cont.

• The motion may be symmetrical– Then tup = tdown

– Then v = -u

• The motion may not be symmetrical– Break the motion into various parts• Generally up and down

Non-symmetrical Free Fall

• Need to divide the motion into segments

• Possibilities include– Upward and downward

portions– The symmetrical portion

back to the release point and then the non-symmetrical portion

Free fall Example #1

• A pebble is dropped down a well and hits the water 1.5 seconds later. Using the equations for motion with constant acceleration, – Determine the distance from the edge of the well

to the water’s surface.

– For the same well, determine how long it would take to hit the water if it was thrown down with a speed of 1 m/s.

Free Fall Example #2

• A person throws a ball upward into the air with an initial velocity of 15.0 m/s. Calculate:– How high it goes– How long the ball is in the air before it comes back

to his hand– How much later the ball hits the ground if he

didn’t catch it, if his hand is 1 meters above the ground.

Free Fall Example #3

• A mass is thrown upwards with an initial velocity of 30 m/s. A second mass is dropped from directly above, a height of 60 m from the first mass, 0.5 s later. – When do the masses meet?

– How high is the point where they meet?

Falling Cats

• Study was done to find out how cats survived long falls.

• Results: The higher the cats were dropped from, the less injuries occurred.

•WHY????

Investigating Air resistance

• Take a few minutes to experiment with dropping some objects and thinking about your everyday experiences.

Forces and Acceleration

• Think of the times when you are accelerating: in the car speeding up, in the car slowing down, on a roller coaster, skiing off a jump.

• Now think of the times when you are not accelerating – travelling at a constant speed in the car, sitting at rest at your desk.

• How do you feel? If your eyes are closed, can you feel whether you are accelerating or not?

Air Resistance and Terminal Velocity

T = 0V =0

T= 1V = 10m/s

T =2 V=20m/s

T =3V=30m/s