Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 24 views

    The DErivatives

    Uploaded by

    Eden Mae P. Cabale
    The document discusses key concepts in the study of motion including displacement, velocity, and acceleration. It provides examples of how these quantities can be represented by functions of time and related through derivatives. Specifically, it shows that: 1) Displacement is the shortest distance between positions and can be represented by a function x(t). 2) Velocity refers to speed and direction, which can be represented by v(t). 3) Acceleration is the rate of change of velocity with respect to time and can be found by taking the derivative of v(t). The document then works through examples calculating displacement, velocity, acceleration, and times when these quantities are at maximum or minimum values. It demonstrates

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd

    The DErivatives

    Uploaded by

    Eden Mae P. Cabale
    24 views17 pages
    The document discusses key concepts in the study of motion including displacement, velocity, and acceleration. It provides examples of how these quantities can be represented by functions of time and related through derivatives. Specifically, it shows that: 1) Displacement is the shortest distance between positions and can be represented by a function x(t). 2) Velocity refers to speed and direction, which can be represented by v(t). 3) Acceleration is the rate of change of velocity with respect to time and can be found by taking the derivative of v(t). The document then works through examples calculating displacement, velocity, acceleration, and times when these quantities are at maximum or minimum values. It demonstrates

    Original Description:

    Copyright

    © © All Rights Reserved

    Available Formats

    PPTX, PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document discusses key concepts in the study of motion including displacement, velocity, and acceleration. It provides examples of how these quantities can be represented by functions of time and related through derivatives. Specifically, it shows that: 1) Displacement is the shortest distance between positions and can be represented by a function x(t). 2) Velocity refers to speed and direction, which can be represented by v(t). 3) Acceleration is the rate of change of velocity with respect to time and can be found by taking the derivative of v(t). The document then works through examples calculating displacement, velocity, acceleration, and times when these quantities are at maximum or minimum values. It demonstrates

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    24 views17 pages

    The DErivatives

    Uploaded by

    Eden Mae P. Cabale
    The document discusses key concepts in the study of motion including displacement, velocity, and acceleration. It provides examples of how these quantities can be represented by functions of time and related through derivatives. Specifically, it shows that: 1) Displacement is the shortest distance between positions and can be represented by a function x(t). 2) Velocity refers to speed and direction, which can be represented by v(t). 3) Acceleration is the rate of change of velocity with respect to time and can be found by taking the derivative of v(t). The document then works through examples calculating displacement, velocity, acceleration, and times when these quantities are at maximum or minimum values. It demonstrates

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 17

    THE

    DERIVATIVES
    OBJECTIVE:
    APPLY THE DERIVATIVE CONCEPT
    IN THE STUDY OF MOTION
    Kinematics is the study of motion and is closely related to
    calculus. Physical quantities describing motion can be
    related to one another by derivatives.
    1. DISPLACEMENT IS THE SHORTEST DISTANCE
    BETWEEN TWO POSITIONS AND HAS A DIRECTION.

    Example:
    • The park is 5 kilometers north of here
    • x(t) =5(t), where x is displacement from a point P
    and t is time in seconds.
    2. VELOCITY REFERS TO THE SPEED AND
    DIRECTION OF AN OBJECT.

    Example:
    • Object moving 5 m/s backwards
    • v(t)= , where v is an object’s velocity and t is time in
    seconds
    3. ACCELERATION IS THE RATE OF CHANGE OF VELOCITY PER UNIT TIME.
    IMAGINE INCREASING YOUR SPEED WHILE DRIVING. ACCELERATION IS
    HOW QUICKLY YOUR SPEED CHANGES EVERY SECOND.

    Example:
    • Increasing speed from 10 m/s to 25 m/s in 5 s results in:

    • a(t)=-t, where a is an object’s acceleration and t is time in seconds


    GIVEN X(T) AS DISPLACEMENT, V(T) AS VELOCITY AND
    A(T) AS ACCELERATION, WE CAN RELATE THE
    FUNCTIONS THROUGH DERIVATIVES.

    a ( t ) = 1 l( t ) = x 11
    ( t )

    • Equivalently, using Leibniz notation:


    •
    ch, d 2 x
    .a(t) = -= ­
    dt dt 2
    X

    The maximum of the


    displacement function,
    x(t), occurs along the
    flat blue line where the
    t
    8 12
    rate of change is zero.
    EXAMPLE 1:
    If a particle is moving in space with a velocity function, v(t)= -2t-8
    where t is seconds and velocity is measured in meters per
    second.
    (a) At what time(s), if any, is the particle at rest?
    (b) What is the acceleration of the particle at t=3 seconds?
    SOLUTION:
    (a)At what time(s), if any, is the • Since negative time is
    particle at rest? impossible, the only time at
    which the particle is at rest is 4
    • (a) If the particle is at rest, v(t)=0 seconds.
    (velocity is zero at rest)
    V
    Solving for t when v(t) = 0:
    SOLUTION:
    (B)WHAT IS THE ACCELERATION OF THE PARTICLE AT
    T=3 SECONDS?

    • First find the function for • Thus, the acceleration at t = 3 s


    acceleration by taking the is 4 m/s2
    derivative of velocity.

    Substitute t = 3 s in the
    acceleration function: a(3) = 2(3) -
    2 = 4 m/s2
    J,.
    Example 2 :
    A soccer ball is kicked into the air so that the path of its flight
    can be modeled by the function, where t is in seconds and x
    is meters above ground:

    𝑥 ( 𝑡 ) =− 4.9 𝑡 2 +9.8 𝑡 +5

    (a)At what time will the ball land?


    (b)How many meters above ground was the ball kicked?
    (c)What is the maximum height the ball will reach and at what time will
    this occur?
    (d)What is the acceleration (with direction) of the ball at t=3 s?
    SOLUTION:
    (A) AT WHAT TIME WILL THE BALL LAND?

    • Since x(t) models height above


    ground, x(t)=0 when the ball hits the
    ground

    • However, t is greater than 0, (since


    time cannot be negative). Thus, the
    ball hits the ground 2.421 seconds
    after being launched.

    -2.421
    (B)HOW MANY METERS ABOVE GROUND WAS THE BALL
    KICKED?

    • The initial height above ground occurs when t = 0. Substitute t = 0


    into x(t):

    • Thus, the ball is thrown from 5 meters above ground.


    (C)WHAT IS THE MAXIMUM HEIGHT THE BALL WILL
    REACH AND AT WHAT TIME WILL THIS OCCUR?

    • Maximum height occurs


    when the first derivative
    equals zero.
    (D)WHAT IS THE ACCELERATION
    (WITH DIRECTION) OF THE BALL
    AT T=3 S?

    • Acceleration is equal to the


    second derivative of
    t
    displacement.
    THANK YOU!!!

    You might also like