© 2010 Pearson Education, Inc. Motion Is Relative Displacement vs. Distance Velocity vs. Speed Velocity: Average and Instantaneous Acceleration Free Fall.

© 2010 Pearson Education, Inc.

• Motion Is Relative• Displacement vs. Distance• Velocity vs. Speed• Velocity: Average and Instantaneous• Acceleration• Free Fall

I recommend writing down items in these yellow boxes

Chapter 2: Motion in One Direction


“Motion Is Relative” – What does this mean?

Motion of objects is always described as relative to something else. For example:

Motion takes place over time and depends upon the frame of reference (precise location of objects in space).


Displacement – (your thoughts?)The length of the straight line drawn from its initial position to the object’s final position. Displacement: the change in position of an object

Δx = xf – xi

Δ = Change

xf = final position (on x-axis)

xi (xo) = initial position (on x-axis)

Δx = Change in position on the x-axis


Displacement

Δx = xf – xiΔx = Change in position on the x-axis

xf xi

Δx = xf – xi = 7.0 cm – 2.0 cm = 5.0 cm


Displacement

Δx = xf – xiΔx = Change in position on the x-axis

xf xi

Δx = xf – xi = 1.2 cm – 6.5 cm = -5.3cm Positive or negative depends on the direction


Displacement

Δy = yf – yi

Δy = Change in position on the y-axisyf

yi

Δy = yf – yi

= 10.0cm – 6.1 cm

= 3.9cm

Positive or negative depends on the direction


Displacement = Distance ???xf xi

xf xi


Displacement = Distance ???

xf xi

xf xi xf

Distance does not have positive or negative, it is the total distance traveled, NOT just the final outcome (xf – xi).


Velocity – (your thoughts?)

NOTE: For most problems, the initial values are going to be zero. We typically start at time zero, then begin. Our displacement also typically starts at a position of zero and moves a certain “distance”.

Δx change in distance Δt change in time

vavg = = = m/s

Δx xf – xi Δt tf – ti

vavg = =dtv = = m/s


Velocity = Speed ??? (your thoughts?)

Average Velocity

Constant Velocity

Instantaneous Velocity


• Velocity (v) is a description of– the instantaneous speed of the object– what direction the object is moving (north, left, right,

up, down, pos x-axis, neg x-axis)

• Vectors have magnitude and direction

• Velocity is a vector quantity. It has– magnitude: instantaneous speed– direction: direction of object’s motion

(typically positive and negative x-axis)

Δx change in distance Δt change in time

v = = = m/sdt

v = = m/s


Speed (s) and Velocity (v)

• Constant speedspeed is steady speed, neither speeding up nor slowing down.

• Constant velocityvelocity is– constant speed and – constant direction (straight-line path with no

acceleration).

Motion is relative to Earth, unless otherwise stated.


Example: A flying disc leave’s Mr. Pearson’s hand with a velocity of 15 m/s. Ignoring wind resistance, the disc travels for 10 seconds before hitting the ground. How far did the disc travel?

Δd change in distance Δt change in time

v = = = m/sdt

v = = m/s

v = 15 m/st = 10 sd = ____ m

d t

v = d = t * v= 10 s * 15m/s

d = 150 m

Ignoring wind resistance = no friction = constant velocity

P.I.E.S.S ???

v = 15 m/s


Velocity Δx xf – xi Δt tf – ti

vavg = =

t0 t1 t2 t3

x0 = 0.0 cmx1 = 1.5 cm x2 = 4 cm x3 = 9 cm

1) What is the displacement between x1 and x2?2) What is the average velocity over the entire

movement from x0 to x3?

3) What is the average velocity between t0 and t1?4) What is the average velocity between x2 and x3?


Δx

Δt

time (s)

ΔxΔt


vavg = =

dis

tan

ce (

m)


time (s)

dis

tan

ce (

m)


vavg = =


(speedometer)




Instantaneous Velocity - is the velocity at any specific instant.

Example: – When you ride in your car, you may speed up

and slow down - (this will affect your average velocity over a given distance and time)

– Your instantaneous speed is given by your speedometer (your speed at that moment).


Average Speed• The entire distance covered divided by the total

travel time– Doesn’t indicate various instantaneous speeds along

the way.

total distance covered

Average speed time interval

Example: What is your average speed if you drive a total distance of 200 km in 2 hours?

200 km2 h

= 100 km/h


The average speed of driving 30 km in 1 hour is the same as the average speed of driving

A. 30 km in 1/2 hour.

B. 30 km in 2 hours.

C. 60 km in 1/2 hour.

D. 60 km in 2 hours.

Average SpeedCHECK YOUR NEIGHBOR


The average speed of driving 30 km in 1 hour is the same as the average speed of driving

A. 30 km in 1/2 hour.B. 30 km in 2 hours. C. 60 km in 1/2 hour.D. 60 km in 2 hours.

Average SpeedCHECK YOUR ANSWER

Explanation:Average speed = total distance / timeSo, average speed = 30 km / 1 h = 30 km/h.

Now, if we drive 60 km in 2 hours:Average speed = 60 km / 2 h = 30 km/h

Same

60 km2 hour

= 30 km1 hour


Acceleration

Formulated by Galileo based on his experiments with inclined planes.


Acceleration (a) - rate at which velocity changes over time

Δv vf - vo change in velocity m/s Δt tf - to time required for change s

a = = = = = m/s2

vf = velocity final

vo = velocity initial (if starting from rest vo = ZERO)

tf = time final

to = time initial

NOTE: Usually we don’t use tf - to just t = required time

a = = = m/s2v m/s t s

vo = velocity at the starting time

At time = 0 sec


Example: • You car’s speed right now is 40 km/h. (vo = 40 km/h)• Your car’s speed 2 s later is 45 km/h. (vf = 45 km/h t = 2s )• Your car’s change in speed is 45 – 40 = 5 km/h. (change = Δv)

Δv vf - vo change in velocity m/s Δt t time required for change s

a = = = = = m/s2

Δv vf - vo 45 km/h – 40 km/h 5 km/h Δt t 2 s 2 s

a = = = = = 2.5 km/h/s

Your car’s acceleration is 2.5 km/h/s.


Δv vf - vo change in velocity m/s Δt t change in time s

a = = = = = m/s2

a = = = m/s2v m/s t s

Example: A flying disc leave’s Mr. Pearson’s hand with a velocity of 15 m/s. After 3 seconds, wind resistance slows down the disc to 11 m/s. What is the acceleration of the disc? ,

vo = 15 m/s

t = 3 s

vf - vo

t

a =

a = 1.3 m/s2

Change in velocity results in Acceleration

vf = 11 m/s11 m/s – 15 m/s

3 s =

-4 m/s3 s

=

The negative means that the acceleration is in the direction opposite to the motion of the object = resulting in slowing



a = = = = = m/s2

a = = = m/s2v m/s t s

Example: If a different disc has an acceleration of -2.5 m/s2. What will the velocity be after 2 seconds of flight if Mr. Pearson releases the disc at 13 m/s.

vf = 8 m/s

Don’t wait for the teacher, get to work

x

Is this answer REASONABLE? Explain


Acceleration

Acceleration involves a change in velocity:

• A change in speed (faster or slower), or

• A change in direction, or

• Both.

When you are driving a car, what are three things you can do to cause acceleration?.

NOTE: You do not feel velocity

You can feel a CHANGE in velocity

You DO feel ACCELERATION


An automobile is accelerating when it isA. slowing down to a stop.

B. rounding a curve at a steady speed.

C. Both of the above.

D. Neither of the above.

AccelerationCHECK YOUR NEIGHBOR


An automobile is accelerating when it isA. slowing down to a stop. B. rounding a curve at a steady speed.C. Both of the above.D. Neither of the above.

AccelerationCHECK YOUR ANSWER

Explanation:• Change in speed (increase or decrease) is acceleration,

so slowing is acceleration.• Change in direction is acceleration (even if speed stays

the same), so rounding a curve is acceleration.


Acceleration and velocity are actuallyA. the same.

B. rates but for different quantities.

C. the same when direction is not a factor.

D. the same when an object is freely falling.

AccelerationCHECK YOUR NEIGHBOR


Acceleration and velocity are actuallyA. the same.

B. rates but for different quantities.

C. the same when direction is not a factor.

D. the same when an object is freely falling.

AccelerationCHECK YOUR ANSWER

Explanation: • Velocity is the rate at which distance changes over time, • Acceleration is the rate at which velocity changes over time.


a = = = = = m/s2


v = = = m/s


AccelerationGalileo increased the inclination of inclined planes.• Steeper inclines gave greater accelerations. • When the incline was vertical, acceleration was

max, same as that of the falling object.• When air resistance was negligible, all objects fell

with the same unchanging acceleration.

Unchanging = CONSTANT Acceleration

Which ball hits the ground first?Which ball has the highest vf ?Which ball has the highest vo ?


Free Fall

Falling under the influence of gravity only - with no air resistance

• Freely falling objects on Earth accelerate at the rate of 10 m/s/s, i.e., 10 m/s2

a = 10 m/s2 = 9.8 m/s2 = 9.81 m/s2 (sig figs)

We say this is the acceleration due to gravity (g):

a = g = 10 m/s2 = 9.8 m/s2

Text book vs Real World(easier math)


Free Fall—How Fast?The velocity acquired by an object starting from rest is

So, under free fall, when acceleration is a = g = 10 m/s2, the speed is

v0 = 0 m/s before release: t0 = 0 s

v1 = 10 m/s after 1 s t1 = 1 s

v2 = 20 m/s after 2 s. t2 = 2 s

v3 = 30 m/s after 3 s. t3 = 3 s

And so on.

Velocity acceleration x time

a = Δv t

Acceleration = Change of 10 m/s per second


Gravity WITH Friction

In this situation g still equals 10 m/s2, but the motion of the object is no longer strait down. Here, we must include friction and trigonometry. In this example: a = 4 m/s2 NOT 10 m/s2

v0 = 0 m/s before release: t0 = 0 s

v1 = 4 m/s after 1 s t1 = 1 s

v2 = 8 m/s after 2 s. t2 = 2 s

v3 = 12 m/s after 3 s. t3 = 3 s

And so on.

Velocity acceleration x time

a = Δv t

Acceleration = Change of 4 m/s per second


A free-falling object has a speed of 30 m/s at one instant. Exactly 1 s later its speed will be

A. the same.

B. 35 m/s.

C. more than 35 m/s

D. 60 m/s.

Free Fall—How Fast?

a = g = 10 m/s2

a = Δv t Δ v = a*t

vf = vo + a*tvf = vo + Δv

vf = 30m/s + (10m/s2 * 1s)

vf = 30m/s + 10m/svf = 40m/s

vo = 30 m/st = 1 svf = ____ m/s

Freely Falling


Free Fall—How Far?The distance covered by an accelerating object -- starting from rest is:

So, under free fall, when acceleration is 10 m/s2, the distance is

• at t = 1 s; d = ½ *a*(t)2 = ½ (10 m/s2) (1 s)2 = 5 m

• at t = 2 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2 s)2 = 20 m

• at t = 3 s; d = ½ *a*(t)2 = ½ (10 m/s2) (3 s)2 = 45 m

And so on

d = ½ *a*(t)2

m = ½ *(m/s2)*(s)2

Note that the seconds cancel


What is the distance covered of a freely falling object starting from rest after 4 s?

A. 4 m

B. 16 m

C. 40 m

D. 80 m

Free Fall—How Far?CHECK YOUR NEIGHBOR

vo = 0 m/s (rest)t = 4 sd = ____ m

a = g = 10 m/s2

d = ½ *a*(t)2

d = ½ *(10 m/s2)*(4 s)2

d = (5 m/s2) * (16 s2)

d = 80 m

Freely Falling


EXAMPLE: A ball thrown straight up into the air begins to slow down (constant acceleration) until it stops in mid air at a height of 34 meters. How long does it take to reach this height? (NOTE: the same equation for free-fall is the same)

d = 34 mt = _____ s

a = g = 10 m/s2

Freely Falling

d = ½ *a*(t)2 2*d = (t)2

at = (2*d/a)1/2

t = (2*34 m / 10m/s2)1/2

t = (68 m / 10m/s2)1/2

t = (6.8 s2)1/2

t = 2.6 s d = 34 m

Is this answer REASONABLE?


Free Fall—How Far?at t = 1 s; d = ½ *a*(t)2 = ½ (10 m/s2) (1 s)2 = 5 m

at t = 2 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2 s)2 = 20 m

d = ½ *a*(t)2

at t = 3 s; d = ½ *a*(t)2 = ½ (10 m/s2) (3 s)2 = 45 m

at t = 2.6 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2.6 s)2 = 34 m


v = = = m/sdt

v = = m/s


a = = = = = m/s2

d = ½ *a*(t)2 = ½ *(m/s2)*(s)2 = m

a = g = 10 m/s2 = 9.8 m/s2 (for freely falling objects)


Figure 3.8

t0 = 0 s v0 = 30 m/s

t1 = 1 s v1 = 20 m/s

t2 = 2 s v2 = 10 m/s

t3 = 3 s v3 = 0 m/s

t4 = 4 s v4 = -10 m/s

t5 = 5 s v5 = -20 m/s

t6 = 6 s v6 = -30 m/s

t7 = 7 s v7 = -40 m/s

(negative = direction)

a = g = 10 m/s2

a = Δv t

vf = vo + a*t

t4 = 1 s

t5 = 2 s

t6 = 3 s

t7 = 4 s

d = ½ *a*(t)2


© 2010 Pearson Education, Inc. Motion Is Relative Displacement vs. Distance Velocity vs. Speed Velocity: Average and Instantaneous Acceleration Free Fall.

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