1D Motion

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# 1D Motion - PowerPoint PPT Presentation

X. 1D Motion. Sept. 2013. Contents. 1D Motion, Kinematics and Dynamics - definition Speed and Velocity Acceleration Equation of Kinematics Freely falling bodies Graphical analysis of velocity and accelration. 1D Motion, Kinematics and Dynamics.

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X

### 1D Motion

Sept. 2013

Contents
• 1D Motion, Kinematics and Dynamics - definition
• Speed and Velocity
• Acceleration
• Equation of Kinematics
• Freely falling bodies
• Graphical analysis of velocity and accelration
1D Motion, Kinematics and Dynamics

One dimensional motion is motion along a straight line, like the motion of a train on a track.

There are two aspects to any motion: The movement itself. Is it rapid or slow, for instance?What causes the motion or what changes it? Apply the breaks in a car

Terms (cont.)

Kinematics – deals with the concepts that are needed to describe motion, without any reference to forces

Dynamics - deals with the effect that forces have on motion

Kinematics and Dynamics form the branch of physics known as Mechanics

Displacement

To describe the motion of an object, we must be able to specify the location of the object at all times

Displacement of the car

is a vector drawn from

the initial position to

the final position.

Displacement is a vector

quantity

Final

Initial

Term - Displacement

Displacement in one direction along the line is assigned a positive value, and a displacement in the opposite direction is assigned a negative value

Displacement does not give information of the location, it gives information about the change in location in specific time.

Average Speed

Usain Bolt run 100 meters in 9.77 seconds last August in Moscow. How fast did he run?

100 m / 9.77 sec = 10.23 m/sec

Average speed is the distance traveled divided by the time required to cover the distance (always positive)

Example

My dog chased a cat in our backyard. It run a distance of 30 meters till the cat was lost. The average speed was 6 m/s. How much time did it take till the cat was gone?

Example (cont.)

= 30 m / 6 (m/sec) = 5 sec

The dog ran 5 sec till it lost the cat.

Average Velocity

Speed indicates how fast an object is moving.

However, speed does not reveal anything about the direction of the motion.

To describe both how fast an object moves and the direction of its motion, we need the vector concept of velocity.

Term – Average Velocity

is the initial location at time

is the final location at time

Average Velocity – (cont.)

Average velocity is a vector that points in the same direction as the displacement

we can represent the direction of motion with a +/– sign

Objects A and B have thesame speed s = |v| = +10 m/s, but they have different velocities

We can rewrite the equation as follows:

We can see that the location is defined by the initial location summed up with the multiplication of the average velocity by the time period

V=3 cm/s

X=X0+Vt

Example

A worm is located 2cm to the right of the origin when measurement start.

It crawls towards the origin at constant velocity of 3 cm/sec

• Write a function that describe the relation between location and time
• Draw a graph of the location at different times
• Draw a graph of the average velocity at different times
• What is the displacement during 10 sec of movement
• What is the distance passed in 10 sec
• What is the displacement during the 4th second of movement

X=X0+Vt

• Write a function that describe the relation between location and time
• First we decide of the positive direction – it will be to the right
• The positive direction impacts the signs of the velocity and the initial location, in this example the velocity will be negative and the location positive

V=-3 cm/s

X(cm)

X0=+2 cm

X=X0+Vt

• Write a function that describe the relation between location and time

= 0

X=X0+Vt

X=2-3t

V=-3 cm/s

X(cm)

X0=+2 cm

X(cm)

t(s)

X=X0+Vt

• Draw a graph of the location at different times

X=2-3t

2

V=-3 cm/s

X(cm)

X0=+2 cm

X=X0+Vt

• Draw a graph of the average velocity at different times

V(cm/s)

t(s)

-3

V=-3 cm/s

X(cm)

X0=+2 cm

X=X0+Vt

• What is the displacement during 10 sec of movement?

X=X0+Vt

∆X=Vt

X-X0=Vt

∆X=-3∙10=-30cm

V=-3 cm/s

X(cm)

X0=+2 cm

X=X0+Vt

• What is the distance passed in 10 sec?

When a body is moving at constant velocity at all times, the distance passed is equal to the absolute value of the displacement

S=|∆X|=|-30|=30cm

V=-3 cm/s

X(cm)

X0=+2 cm

X=X0+Vt

• What is the displacement during the 4th second of movement?

The 4th second of movement, is 1 second long as any other second.

Constant velocity motion has constant displacement at any second

-3∙1=-3cm=∆X=Vt

V=-3 cm/s

X(cm)

X0=+2 cm

Instantaneous Velocity

The magnitude of average velocity is an average, hence does not convey any information about how fast you were moving or the direction of the motion at any instant during the trip

The instantaneous velocity of the car indicates how fast the car moves and the direction of the motion at each instant of time

The magnitude of the instantaneous velocity is called the instantaneous speed

Instantaneous Velocity (cont.)

The notation means that the ratio is defined by a limiting process. Smaller and smaller values of t are used, so small that they approach zero.

As smaller t is used, x also becomes smaller.

However, the ratio does not become zero,it approaches the value of the instantaneous velocity.

Acceleration

In a wide range of motions, the velocity changes from moment to moment. To describe the manner in which it changes, the concept of acceleration is needed

The change in velocity may occur over a short or a long time interval

Acceleration (cont.)

If the velocity is changing, then there is non-zero acceleration

Rewriting the equation:

We received a new function that describes the velocity of a body at any time t, that started off with velocity Vo and travels at constant acceleration a during the period of time t.

V(m/s)

V0

t(s)

ΔV

Δt

Graph of motion in constant acceleration

The v vs. t graph slope is the acceleration

V(m/s)

V0

ΔV

Δt

t(s)

Acceleration can be Positive or Negative

Negative slope represents negative acceleration

the object slows down and is said to be “decelerating”

Is it true to state that any time a body is decelerating, it is also slowing down?

V(m/s)

t(s)

ΔV

-V0

Δt

Negative slope, represent negative acceleration

Motion is in negative direction, hence velocity is negative.

Attention!!! The magnitude of the velocity is increasing

+

X

Is it true to state that any time a body have positive accelerating, it is also speeding up?

Velocity magnitude is decreasing, acceleration is positive

Velocity is negative because motion is

towards negative X,

and in the same time the velocity is decreasing

V(m/s)

t(s)

Δt

ΔV

Positive Slope

-V0

+

X

V(m/s)

ΔV

V0

Δt

t(s)

If velocity is increased while moving in positive direction, both velocity and acceleration are positive

+

X

V(m/s)

t(s)

ΔV

V0

Negative Slope =>

Negative acceleration

Δt

If velocity is increased while moving in negative direction, both velocity and acceleration are negative

Velocity increased to the left

a

Const. Acceleration left

+

X

V(m/s)

V0

ΔV

Δt

If velocity is decreased while moving in positive direction, the velocity is positive, and the acceleration is negative

t(s)

Velocity decreased to the right

Const. Acceleration to left

+

X

V(m/s)

t(s)

Δt

ΔV

V0

If velocity is decreased while moving in negative direction, the velocity is negative, and the acceleration is positive

V

Velocity decreased to the left

Const. Acceleration to right

a

+

X

Const. negative acceleration

V(m/s)

V0

Velocity magnitude decrease

Velocity decrease

ΔV

Δt

t(s)

Temp stop to switch motion direction

Velocity increase

Velocity magnitude increase

+

X

Const. positive acceleration

V(m/s)

Velocity magnitude increase

ΔV

Velocity Increase

Δt

t(s)

Temp stop to switch motion direction

Velocity decrease

-V0

Velocity magnitude decrease

+

X

Instantaneous Acceleration

Instantaneous Acceleration is an object’s acceleration at a particular instant of time

Instantaneous acceleration is a limiting case of the average

acceleration.

When the time interval becomes extremely small (approaching zero in the limit), the average acceleration and the instantaneous acceleration become equal

Example 1

The plane in the figure starts from rest (Vo = 0 m/s) when t0 = 0 s. The plane accelerates down the runway and at t=29 s attains a velocity of V=260 km/h, where the plus sign indicates that the velocity points to the right.

Determine the average acceleration of the plane?

Solution 1

The average acceleration of the plane is defined as the change in its velocity divided by the elapsed time

Assuming the acceleration of the plane is constant, a value of “nine kilometers per hour per second” means the velocity changes by +9.0 km/h during each second of the motion

Solution 1- cont.

It is customary to express the units for acceleration solely in terms of SI units.

convert the velocity units from km/h to m/s :

The average acceleration then becomes

The velocity changes by 2.5 m/s during each second of the motion

Example 2

A drag racer crosses the finish line, and the driver deploys a parachute and applies the brakes to slow down.

The driver begins slowing down when t0=9.0 s and the

car’s velocity is V0=28 m/s. When t=12.0 s, the velocity has been reduced to V=13 m/s. What is the average acceleration of the dragster?

Solution 2

The average acceleration of an object is always specified as its change in velocity,V-V0 , divided by the elapsed time, t-t0.

This is true whether the final velocity is less than the initial velocity or greater than the initial velocity

Equation of Kinematics (a = const.)

We already saw earlier the equation of velocity:

V = V0 + at

If X0 = 0 and t0 = 0 we reduce the average velocity equation to:

Equation of Kinematics (a = const.)

Because the acceleration is constant, the velocity increases at a constant rate. Thus, the average velocity is midway between the initial and final velocities

And we get the following functions:

Equation of Kinematics (a = const.)

If t0 = 0 then we get

Arranging the equation:

And finally:

Solution

Using equation:

Since the spacecraft is slowing down, the acceleration must be opposite to the velocity

Both of these answers correspond to the same displacement (x= +215 km), but each arises in a

different part of the motion

Freely falling bodies

It’s well known that gravity causes objects to fall downward.

In the absence of air resistance, all bodies at the same location above the earth fall vertically with the same acceleration.

If the distance of the fall is small compared to the radius of the earth, the acceleration remains essentially constant.

This idealized motion, in which air resistance is neglected and the acceleration is nearly constant, is known as free-fall

Free fall on the moon

Here's the famous footage of the Apollo 15 astronaut that dropped a hammer & feather on the moon to prove Galileo's theory that in the absence of atmosphere, objects will fall at the same rate regardless of mass

The acceleration due to gravity near the surface of the moon is approximately one-sixth as large as that on the earth

Term – Acceleration due to gravity

The acceleration of a freely falling body is called the acceleration due to gravity.

It’s magnitude is denoted by the symbol g.

It’s directed downward, toward the center of the earth.

Near the earth’s surface, g is approximately

When the equations of kinematics are applied to free-fall motion, it is natural to use the symbol y for the displacement, since the motion occurs in the vertical or y direction

1Example

A stone is dropped from rest

from the top of a tall building.

After 3.00 sec of free-fall, what

is the displacement y of the

stone?

Solution 1

Check Equation

Example 1 – cont.

After 3.00 sec of free-fall, what is the velocity v of the stone?

Check Equation

Important!

The acceleration due to gravity is always a downward-pointing vector. It describes how the speed increases for an object that is falling freely downward.

This same acceleration also describes how the speed decreases for an object moving upward under the influence of gravity alone, in which case the object eventually comes to a momentary halt and then falls back to earth

Example 2

A football game customarily begins with a coin toss to determine who kicks off. The referee tosses the coin up with an initial speed of 5.00 m/s. In the absence of air resistance,how high does the coin go above its point of release?

Solution 2

Upward initial velocity, but the acceleration due to gravity points downward.

Since the velocity and acceleration point in opposite directions, the coin slows down as it moves upward.

Eventually, the velocity of the coin becomes v=0 m/s at the highest point

Check Equation

Example 2 – cont.

What is the total time the coin is in the air before returning to its release point?

Solution 3

T1

T2

The motion of an object that is thrown upward and eventually returns to earth has a symmetrythat is useful to keep in mind from the point of view of problem solving.

The calculations just completed indicate that a time symmetry exists in free-fall motion, in the sense that the time required for the object to reach maximum height equals the time for it to return to its starting point (T1 = T2).

A type of symmetry involving the speed also exists.

Solution 3 – cont.

If the pellet was shoot straight up with velocity +30 m/sec, than from symmetry we can tell that when it will be back at the initial height , it will have same velocity but in opposite direction.

Hence the pellet will hit the ground in the same velocity on both cases.

Summary

Topics covered:

Units and dimensions

Unit conversion

Linear functions and graph

Trigonometry

Next meeting:

Scalars and vectors