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Physics 1110: 1D Kinematics

•  CAPA homework due tomorrow night!

•  Second CAPA Assignment due Friday night.

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

Announcements:

•  This weeks tutorial (pages HW 3-6 ) is due at your recitation at start of recitation.

Today we will finish chapter 2 of H+R

New Assignment Calendar on Webpage

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A. Both trains speed up all the time. B: At time tC, both trains have the same velocity. C: Both trains have the same velocity at some time before tC. D: At some time, both trains have the same acceleration. E: 0 or 2 of the above statements are true.

Clicker question 1 Q. The graph show the positions as a function of time for two trains running on parallel tracks. Which of the following statements is true?

Set frequency to BA

time

position train A

train B

tC

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Clicker question 2 Q. An object’s position vs time graph is shown. What best describes the car’s velocity vs time?

Set frequency to BA

E. None of these

x

t

t

v A t

v

t

v

t

v D

B

C

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Recap from Friday Displacement is

Average velocity is so

Instantaneous velocity is

Average acceleration is

Instantaneous acceleration is

for constant velocity

for constant acceleration

v(m/s)

t(s) 0 10

20 In this example,

and

Average velocity with constant acceleration

“Proof” that when acceleration is constant

Let us take an average of equally spaced points to obtain the average

Avg for t=0,5,10:

Avg for t=0,2,4,6,8,10:

Can divide into smaller and smaller pieces until eventually you have infinitesimal pieces to measure the average

Particle spends equal amounts of time below & above 10 m/s

Now do it more rigorously

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Average velocity with constant acceleration

“Proof” that when acceleration is constant,

€

v =vdt

0

t∫dt

0

t∫

=(v0 + at)dt0

t∫

t=(v0t +

12at 2)

t

v = v0 +12at = v0 + (v f − v0) /2

v =v0 + v f2

€

v f = v0 + at

€

at = v f − v0or

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Relation between velocity & acceleration

v(m/s)

t(s) 0

v(m/s)

t(s) 0

t(s)

v(m/s)

0

Examples of constant acceleration

Velocity starts at 0 and increases. Acceleration > 0

Velocity starts at 0 and decreases. Acceleration < 0

Velocity starts positive and decreases to 0 and then decreases more. Acceleration < 0

Acceleration is the change in velocity, not the velocity

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Clicker question 3 Set frequency to BA

A glider on a tilted air track is given a brief push uphill. The glider coasts up to near the top end, stops, and then slides back down. When the glider is at the highest point of its path, its acceleration is..

A) straight down B) downward along the track C) upward along the track D) no direction, the acceleration is zero.

The direction of the acceleration vector is the direction of the Δv vector

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Constant acceleration For constant acceleration: Setting t0 = 0 :

Remember:

For constant acceleration:

(for constant acceleration only)

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Constant acceleration (continued) Does make sense?

We know which gives

We know which gives

For zero acceleration we get which is what we obtained for the constant velocity case

Seems to work!

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Another constant acceleration formula (always true)

(constant acceleration)

Note that the acceleration is absent from this equation

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Yet another constant acceleration formula

Solving for gives

Substitute into

to get

Rearrange terms to find

Note that the time is absent from this equation

Subtract and multiply by to get

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Gravity The most obvious form of constant acceleration is from the force of gravity. On the surface of the Earth gravity causes a downward acceleration of magnitude 9.8 m/s2 which is labeled “g”:

Aristotle believed that heavier objects fell faster than lighter objects

Galileo figured out that all objects are subject to the same acceleration, first by dropping balls from the Tower of Pisa and later by rolling balls down inclines

Gravity 0.5% higher at poles than equator and 0.2% higher at Dead Sea (lowest) than Mt Everest (highest)

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A.  zero B.  straight up C.  straight down D.  depends on the ball mass

Clicker question 4 Q. A ball is thrown straight up. At the top of its trajectory, its acceleration is:

Set frequency to BA

• At the top, the velocity is zero • The acceleration due to gravity is constant; it is always straight down

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Clicker question 5 Set frequency to BA

+y

0

h

v0=0

A rock is dropped from rest and falls a distance h (h > 0 ) to the ground. Down is chosen as the positive direction and the origin is placed at ground level. What is y0 (the initial position) and what is a (the acceleration)?

A) y0 = +h , a = – g B) y0 = –h , a = + g C) y0 = 0 , a = + g D) y0 = – h , a = – g E) y0 = 0 , a = – g

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Summary of constant acceleration equations

No displacement in this equation

No final velocity in this equation

No time in this equation

No acceleration in this equation

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Clicker question 6 Set frequency to BA

A rock is thrown downward with an initial speed |v0| from the edge of a cliff. Assume no air resistance. It falls straight down and strikes the ground after falling a distance h. A student is asked to compute the final speed of the rock, just before it hits the ground. Which one formula should she use?

A) v=v0 + a t B) x=x0 + v0 t +(1/2) a t2 C) v2 = v0

2 + 2 a (x –x0) D) None of these will work E) More than one of these will work OK.

Summary •  Check the web site calendar. •  Clickers counted today!

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