acceleration

Simplest case a=constant. Equations hold even if Δt large.Δv =vf -vi

tavt

va

atvv if

ti= 0

Example : If a car traveling at 28 m/s is brought to a full stop 4.0 s after the brakes are applied, find the average acceleration during braking.

2av m/s 0.7

s 0.4

m/s 280

t

va

Given: vi = +28 m/s, vf = 0 m/s, and t = 4.0 s.

If a = const.

)(2

1fiav vvv

Not true in general

If a = const. one dimensional motion

Fig. 04.01

Vav gives

same areaHence same distance

Constant acceleration

tavv

tavvv

iav

iiav

21

2

)(

atvv if

)(2

1fiav vvv

Δx= vavt

Δx= = vit+1/2 at2

ti = 0

Fig. 04.02

Δx=vxΔt=Area

=

viΔt (blue area)+½ Δt (aΔt) (gold area)

=

viΔt + ½ aΔt2

a=constant • Δx=vavΔt = 1/2(vi+vf)Δt

but vf = vi + aΔt so Δt = (vf-vi)/aΔx = 1/2(vi+vf)Δt = 1/2 (vi+vf) (vf-vi)/a

• = 1/(2a) (vf2-vi

2) = Δx

• (vf2-vi

2) =2aΔx

Plotted vs time

Minimum length of runway

A fully loaded 747 with all engines at full throttle accelerates at 2.6 m/s2. Its minimum takeoff speed is 70 m/s. How long will it require to reach take off speed? What is the minimum length of a runway for a 747.

(vf2-vi

2) =2aΔx

vf = vi + aΔt

Problem

• A car traveling at a speed of 30 m/s. A deer runs across the road and the driver slams on the brakes. It takes .75 s to begin applying the brakes. With the brakes on the car decelerates at 6 m/s2. How far does the car travel from the instant the driver sees the deer until he stops.

Free Fall• All objects, under the influence of only gravity

fall. (We are neglecting air resistance)• They all fall with a constant acceleration

(down) of • g = 9.8 m/s2

• The mass of the object doesn’t matter! Heavy and light objects all fall with the same g

• It doesn’t matter in which direction it is moving it has an acceleration of g

• Since we normally take y + up free fall is -g

Free Fall

Slide 2-36

• You drop a stone off a cliff and hear it hit the ground after two seconds. How high is the cliff?

• This is an how far question• Δx=1/2 at2

• Substitute the numbers a=9.8 t=2• Δx=19.6• How fast is it going when it hits?

Tennis balls are tested by measuring their bounce when dropped from a height of approximately 2.5 m. What is the final speed of a ball dropped from this height?

Example Problem

Slide 2-34

Throwing stones (up)

• What happens if you toss a stone straight up?

v(3)

v(1)

v(0)

. v(4) =0. It reaches its highest point when it stops going up, i.e. when v = 0

a is always downward it is g

Throwing stones (up)• What happens when it starts coming down

again?

v(3) v(1)

v(0)

. v(4) =0. It reaches its highest point when it stops going up and then begins to fall with a=-g. It re-traces its path and velocity but down

a is always downward it is g

An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which choice below best represents the arrow’s acceleration at the different points?

Checking Understanding

A. A E B D; C 0

B. E D C B A

C. A B C D E

D. A B D E; C 0

Slide 2-37

An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which choice below best represents the arrow’s acceleration at the different points?

Answer

A. A E B D; C 0

B. E D C B A

C. A B C D E

D. A B D E; C 0

Slide 2-38

An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

Checking Understanding

Slide 2-39

An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

Answer

D.

Slide 2-40

Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise?

Given: viy = +15.0 m/s; ay = 9.8 m/s2

x

yviy

ay

• Also an how far question

• (vf2-vi

2) =2ad • What is vf? vi? What is a? (be careful of signs!)

• You toss a ball straight up with an initial vi=25m/s. You then become distracted. How long until the ball clunks you on your head?

Example : A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. How long will it take to hit?

369 m

x

y

Given: viy = 0 m/s, ay = 9.8 m/s2, y = 369 m

Unknown: vyf

ay