0 5 10 15 20 25 30 35 40 450

20406080

100120140160180200

Velocity vs Time

Time (s)V

elo

city

(m

/s)

Analysis of a position vs. time graph

Analysis of a velocity vs. time graph0 5 10 15 20 25 30 35 40 45

050

100150200250300350400450500

Position vs Time

Time (s)

Po

sitio

n (

m)

What can be determined from a position vs. time graph?

What can be determined from a velocity vs. time graph?

The slope of the a tangent line to the curve at a point will provide us with instantaneous velocity information.

The slope of a line connecting the initial and final point will provide average velocity information.

The slope of the a tangent line to the curve at a point will provide us with instantaneous velocity information.

The slope of line connecting the initial and final point will provide average velocity information.

vt

xSlope

vdt

dxSlope

at

vSlope

adt

dvSlope

Note that the average and instantaneous velocities are the same. This is only because the acceleration is constant!!

Note the difference between the two lines!

bmxy Equation of a line.

Integration is the opposite of differentiation. Differentiation breaks up a function into its constituent parts, while an integral combines all the parts into a whole.

1 nn nxxdx

d 1

1

1

nn xn

dxx

Example:

cbtattx 3

dt

txdtv

txdt

dtv

cbtatdt

dtv 3

23btatv

→𝑑𝑥=𝑣𝑥𝑑𝑡dt

dxvx

→𝑥0

𝑥

𝑑𝑥=𝑡 0

𝑡

𝑣𝑥𝑑𝑡

→𝑥0

𝑥

𝑑𝑥=𝑡 0

𝑡

(𝑎+3𝑏𝑡 2 )𝑑𝑡

→ [𝑥 ] 𝑥0𝑥 =[𝑎𝑡+𝑏𝑡 3 ]𝑡 0

𝑡

→𝑥−𝑥0=[𝑎𝑡+𝑏𝑡 3 ]− [𝑎𝑡0+𝑏𝑡 03 ]

This is a constant we will call c

→𝑥= [𝑎𝑡+𝑏𝑡 3 ]+𝑥0− [𝑎𝑡 0+𝑏𝑡03 ]

→𝑥=𝑎𝑡+𝑏𝑡 3+𝑐

A train car moves along a long straight track. The graph shows the position as a function of time for this train. The graph shows that the train:

1. speeds up all the time.2. slows down all the time.3. speeds up part of the time and slows down part of the time.4. moves at a constant velocity.

The slope of the curve (or slope of the tangent line to the curve) decreases in time.

The graph shows position as a function of time for two trains running on parallel tracks. Which is true?

1. At time tB, both trains have the same velocity.2. Both trains speed up all the time.3. Both trains have the same velocity at some time before tB.4. Somewhere on the graph, both trains have the same acceleration.

The slopes of the two curves (or the tangent lines to the curves) are parallel at some point prior to tB.

Freefall – a specific example of 1D motion

What is freefall?Motion towards the Earth only due to gravitational attraction.

Close to the surface of the Earth it is commonly discussed as vertical motion.

Gravitational attraction causes an object to be accelerated towards the surface of the Earth.

Near the surface of the Earth this acceleration is assumed to be constant. It is not actually constant it changes with altitude and latitude.

2281.980665.9s

m

s

mgag

• If we assume up is a positive direction the direction of g would be negative. • You can define down as a positive direction. • Make sure you specify your positive coordinate directions.• Once you choose your directions for a specific problem do not change them!!

0

1

2

3

4

5

A ball is thrown upward with an initial speed of 20 m/s.1. What is the velocity, and acceleration at point 0?

2. What is its velocity and acceleration at point 1 after 1 s?

3. What is its velocity and acceleration at point 2?

4. What is its velocity and acceleration at point 3, 1 s after point 2?

5. What is its velocity and acceleration at point 4?

6. What is its velocity and acceleration at point 5, 3 s after point 2?

v = 20 m/s and a = -9.8 m/s2

v = 10.2 m/s and a = -9.8 m/s2

v = 0 m/s and a = -9.8 m/s2

v = -9.8 m/s and a = -9.8 m/s2

v = -20 m/s and a = -9.8 m/s2

v = -29.4 m/s and a = -9.8 m/s2