Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Lecture 3

Chapter 2

Equations of motion for constant acceleration

Physics I

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Today we are going to discuss:

Chapter 2:

Motion with constant acceleration: Section 2.4 Free fall (gravity): Section 2.5

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Finding Position from a Velocity Graph

The total displacement ∆s is called the “area under the velocity curve.” (the total area enclosed between the t-axis and the velocity curve).

∙ Let’s integrate it: ∙ x ∙

∙

Initial position

Geometrical meaning of an integral is an area

Total displacement

displacement

v(t)

tfiif tandtbetweentvsvunderAreaxx

Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s.At t = 4.0 s, the object’s position is

A) 20 m

B) 16 m

C) 12 m

D) 8 m

E) 4 m

ConcepTest Position from velocity

Displacement = area under the curve

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Simplifications

Objects are point masses: have mass, no size

In a straight line: one dimension

Point mass

Consider a special, important type of motion:

Acceleration is constant (a = const)

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

The Kinematic Equations of Constant Acceleration

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Velocity equation. Equation 1.(constant acceleration)

0)(, 00

tandtt

vtvaonacceleratidefinitionby o

t

vtva o)(

the velocity is increasing at a constant rate

v0

v

t

atvtv o )( (1)

Velocity equation

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

tvtva o

)(

ff tandtbetweentvsvunderAreaxx 00

v0

v

tO

A

B

CD

Recall Eq (2.11)

vf

t0=0

ABDOADCf AAxx 0

OADCA

ABDA tvvtvxx ff )( 021

00 atvtv o )(

221

00 attvxx f

Position equation. Equation 2 (constant acceleration)

(2)

Position equation

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

We can also combine these two equations so as to eliminate t:

No time equation. Equation 3 (constant acceleration)

It’s useful when time information is not given.

221

00 attvxx f

atvtv o )((3)

No time equation

Position equation

Velocity equation

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

We now have all the equations we need to solve constant-acceleration problems.

Motion at Constant Acceleration (all equations)

221

00 attvxx f

atvtv o )(

(3)

No time equation

Position equation

Velocity equation

(2)

(1)

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

How to solve:– Divide problem into “knowns” and “unknowns”– Determine best equation to solve the problem– Input numbers

Problem Solving

ExampleA plane, taking off from rest, needs to achieve a speedof 28 m/s in order to take off. If the acceleration of theplane is constant at 2 m/s2, what is the minimum lengthof the runway which can be used?

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Exa

mpl

e

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Free

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Freely Falling Objects

Near the surface of the Earth, all objects experience approximately

the same acceleration due to gravity.

All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s2

Air resistance is neglected

One of the most common examples of motion with constant acceleration is freely falling objects.

The ball is dropped from rest, so its initial velocity is zero. Because the y-axis is pointing upward and the ball is falling downward,its velocity is negative and becomes more and more negative

as it accelerates downward.

You drop a ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up).

ConcepTest Free Fall

v

tA t

B

v

tC

v

tD

yv

v

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Freely Falling Objects

y

g

y

g

a = ‐g

a = g

if then

if then+

++

2

221

00 attvxx f

(3)No time equation

Position equationatvtv o )(

Velocity equation

(2)

(1)

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Example: Ball thrown upward.

A person throws a ball upward into the air with an initial velocity of 10.0 m/s. Calculate

---------------------------------(a) how high it goes, and (b) how long the ball is in the air before it comes back to the hand. (Ignore air resistance.)

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Example

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Example

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

t (s)

Y (m)

0.4 0.8 1.2 1.6 2

5

3

1

v (m/s)

t (s)

Example

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Thank youSee you on Monday

A cart speeds up toward the origin. What do the position and velocity graphs look like?

ConcepTest 1 Roller Coaster

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Velocity/Acceleration/Position

• 4,5 – negative acceleration, but from 0<t <t4 or t5 – deccelerationbut for t> t4 or t5 – acceleration

a

t

a1>0a2>0a3=0

a4<0a5<0

x0

v

t

12

3

45

v0

t4t5

t

x

t4t5

1 2 3

4

5

V=0

U‐turn

Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov

Determining the Sign of the Position, Velocity, and Acceleration