Motion - WordPress.com · 2020. 7. 5. · car to stop. As you brake, your velocity decreases at a constant rate of 5 m/s . What is the car's stopping distance if your final velocity

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MotionOutline- Fundamental Concepts- Motion in One-Dimension- Motion in Two-Diversions (in a plane)- Relative Motion

Fundamental Concepts

Positron vector: the location of an object with respect to the origin

Displacement Vector: Difference between the final and initial positions of an obtect

Acceleration: Instant changes in the velocity vector of an object

Velocity Vector: Instant changes in position of an object

Position and Displacement Vectors

→ initial position

→ final position

→ displacement

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Motion in One-Dimension

In One-dimensional motion, an object moves on a line. If the line is straight, we call the motion "linear motion".

→ initial position

→ final position

→ displacement

- If the object moves from to in time

- Average velocity is a vector quantity, and it is in the same direction as

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- The average velocity is equal to the instant velocity, if the object moves with a constant velocity.

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Example 1

- Positron charges from 0 to 60 meter in the first second

- The position does not change between t=1 s and t=2 s

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- Position changes from 60 m to -40 m between t=2s and t =3 s

- Position changes from -40 m to 0 m between t=3 s and t=5 s:

- Positron charges from 0 m to 0 m between t=0 and t =5 s:

What if the slope is not constant?

slope is changing.

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Instant velocity

If the displacement vector of an object is a function of time, then the velocity vector at an arbitrary time can be calculated as

Average velocity is always constant, but it is not equal to the instant velocity, if the object is not moving with a constant velocity.

* Instant charges in the velocity is called acceleration

Equations of motion:

→ initial position

→ initial velocity

constant

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Vertical Motion

If an object moves along the vertical axis (y): - It moves under the gravitational force - It accelerates due to the gravity

→ constant acceleration

→ vertical velocity

→ vertical position

If an object is held at some height ( ), and it is dropped at)t=0, it starts falling, and its initial velocity will be zero (

This motion is called free falling.

ExampleIn a 100 meter race an athlete constantly accelerates to 10 m/s during the first four seconds, then she keeps running with a constant velocity for the next four seconds. Before she finishes the race, she constantly decelerates, during 4.7 seconds. Ifthe athlete completes the race in 12.7 seconds, what is her accelerations in the

a) first four seconds,

b) second four seconds,c) last 4. 7 seconds.

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Solution:decelerate → slowing down (the acceleration is negative)

a) In the first four seconds:a

b) The second four seconds:

Athlete runs with a constant velocity

c) The athlete decelerates (a < 0)

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ExampleOn a highway at night you see a stalled vehicle and brake your car to stop. As you brake, your velocity decreases at a constant rate of 5 m/s . What is the car's stopping distance if your final velocity is 30 m/s?

Solution: Equations of motion

ExampleIf the position of an object is given bywhere x is given in meter, and t in second.

a) find the velocity as a function of time. Does the object ever stop?

b) find the acceleration as a function of time.

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Solution

If its velocity becomes zero, the object stops at that time.

a)

b)

Example:

In a crash test a car is moving with 100km/h and it hits an immovable concrete wall. After the crash, the car smashes by 0.75 m and then it completely stops. What is the acceleration of the car during the crash?

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Solution:After the car hits the wall, it slows down and then stops. Meanwhile it moves 0.75 m.

Time-independent equation of motion can be used here.

Motion in a Plane

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Average velocity Instant velocity

Acceleration

Magnitudes

We can decompose a two-dimensional motion into two one-dimensional motions.

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Horizontal Vertical

Vector Notation

Horizontal and the vertical motions have the common time parameter

Projectile Motion→ Launch angle

→ initial velocity

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Horizontal and vertical components of the projectile motion can be written as follows:

Horizontal Vertical

We can also write time-independent equation of motion for each component. However, Since there is no horizontal acceleration in the projectile motion, it will be meaningful only for the vertical component.

Some characteristics of the projectile motion can be summarized as follows:

1- Maximum height

When the object reaches up to the maximum height, its velocity is completely horizontal at this point.

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Using time-independent equation motion

2- Flight timeFlight time is the duration which the motion takes from the beginning to the end. If the projectile motion is symmetric with respect to

→ flight time

3- RangeThe furthest horizontal distance from the launch point

If the projectile motion is symmetric

Range is maximized when

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Example

A foot ball is kicked at an angle with a velocity of20 m/s. Calculatea) The maximum height,b) The time of flight before the ball hits ground,c) The range of the ball,d) The velocity of the ball when it hits ground.

Solution:

- It is a projectile motion- It is symmetric with respect to the maximum height

At the maximum height, the vertical velocity will be zero.a)

First let us see what we know about the motion:

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b) Since the motion is symmetric

c)

d) When it hits ground, and its velocity can becalculated by using the equation of motion for the velocities.

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Example:

A movie stunt driver on a motorcycle speed horizontally off a 50.0 m high cliff. How fast must the motorcycle leave the cliff top to hand ground at a point at which the cameras are away of 90 meter from the base of the cliff?

SolutionFirst let us see what we know about the motion:

- It is half a projectile motion- It is NOT symmetric- Range = 90 meter- Height = 50 meter- Origin is at the cliff top.

→ first we need to calculate the flight time

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Uniform Circular Motion- Trajectory is a circle

→ initial position

→ final position

→ uniform motion

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acceleration

Even though the magnitude of the velocity (speed) is constant, there is still an acceleration (a) , since the velocity vector

——> Speed remains constant, velocity changes direction

The circular motion is a periodic motion, since the motion repeats itself after the object comes backto its initial position. The time to complete one turn is called “period".

Period (T) : unit is second (SI) :dimension is Time

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Number of complete turns per unit time is called frequency (f). If the object moves one complete turn in one period:

frequency: unit is 1/s (Hertz) :Dimension is 1/T

Angular Velocity→ angle cleared in Δt

Angular velocity

Angular velocity

Tangential velocity

When the object completes one turn:

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Example:A stunt pilot executes a uniform circular motion with an airplane. The initial velocity is given by v0= 2500i + 3000j. One minute later its velocity is observed as v=-2500i + 3000j. Find the acceleration in the circular motion.

Solution:→ cleared angle in one minute

In one minute the plane moves half a circle, thus it completes a circle in two minutesT= 2 minutes=120 seconds

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Relative MotionA motion can be observed differently by different observers. This is called relative motion. Relative motion can be understood with coordinate transformations. Let us assume there are two observers, and each observer has its own coordinate system. The coordinate system of an observer is called “reference frame".

S, S' —> reference frames of the observers.

Let us assume that S and S' have the same origin (the observers are at the same point) in the beginning. Let us also assume that S' is moving along the positive x-axis (for simplicity). Δt seconds later their origins will not be at the same point.

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Position of A in S and S'?

→ position in S

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Two observers define the position of A with different position vectors.

transformation of the position vector from S to S’

Now let us see how the observers see if the object at Point A is moving. Assuming the observer S measures its velocity as and S' measures as

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relative velocity

Here we assume S' to move with a constant velocity (v0 = constant). Now, let us take another step and assume that the object is moving onith an acceleration.

acceleration observed in S

acceleration observed in S'

If S' moves with a constant velocity, two observers measure the same acceleration. Such reference frames are called "inertial reference frame".

Galilean transformations

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Galilean transformation for velocities

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Example:

A plane is travelling at a velocity of 100 km/h in negative x-direction. Meanwhile the wind is flowing in negative y-direction at a rate 25 km/h. What is the resultant velocity of the plane velocity?

Solution→ plane velocity

→ wind velocity

observe the result velocity

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Example:A boat is travelling across a river that flows in positive ×-direction at 8.50 m/s. Relative to the water the boat is traveling straight at 11.2 m/s. How fast and which way is the boat moving relative to the banks of the river?

Solution: