Ph 201
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PH 201. Dr. Cecilia Vogel Lecture 3. REVIEW. Motion in 1-D instantaneous velocity and speed acceleration. OUTLINE. Graphs Constant acceleration x vs t, v vs t, v vs x Vectors notation magnitude and direction. Sign of Acceleration. Mathematically

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PH 201

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Ph 201

PH 201

Dr. Cecilia Vogel

Lecture 3


Ph 201

REVIEW

  • Motion in 1-D

    • instantaneous velocity and speed

    • acceleration

OUTLINE

  • Graphs

  • Constant acceleration

    • x vs t, v vs t, v vs x

  • Vectors

    • notation

    • magnitude and direction


Sign of acceleration

Sign of Acceleration

  • Mathematically

    • If (signed) velocity increases, a is +

    • If (signed) velocity decreases, a is -

  • Memorize

    • If velocity and acceleration are in same direction, object will speed up

    • If velocity and acceleration are in opposite directions, object will slow down

  • Physical intuition

    • positive acceleration produced by push or pull in + direction

    • negative acceleration produced by push or pull in - direction


Position velocity acceleration

Position, Velocity, Acceleration

  • Velocity is

    • slope of tangent line on an x vs t graph

    • limit of Dx/Dt as Dt goes to zero

    • the derivative of x with respect to time

    • dx/dt

  • Similarly acceleration is

    • slope of tangent line on a v vs t graph

    • limit of Dv/Dt as Dt goes to zero

    • the derivative of v with respect to time

    • dv/dt

  • If you have position as a function of time, x(t)

    • can take derivative to find v(t)

    • take derivative again to find a(t)


Derivatives of polynomials

Derivatives of Polynomials

  • The derivative with respect to time of a power of t, if C is a constant:

  • Special case, if the power is zero:

  • The derivative of a sum is sum of derivatives:

  • ex


Example

Example

  • ex

  • The acceleration at t=0 is -6 m/s2, and at t=3 is 90 m/s2.

  • The average acceleration between t=0 and t=3 is 39 m/s2


Special case constant velocity

Special Case: Constant Velocity

  • Acceleration is zero

  • Graph of x vs. t is linear

    • slope is constant

  • Average velocity is equal to the constant velocity value, v

becomes

if initial time is zero, and we drop subscript on final variables.


Special case constant acceleration

Special Case: Constant Acceleration

  • If object’s acceleration has a constant value, a,

  • then its velocity changes at a constant rate:

  • And its position changes quadratically with time:


Ph 201

Position with Constant Acceleration

  • Slope of the position graph (velocity) is constantly changing

  • quadratic function of time.


Example1

Example

A little red wagon is rolling in the positive direction with an initial speed of 5.0 m/s. A child grabs the handle and pulls, giving it a constant acceleration of 1.1 m/s2 opposite its initial motion. Let the time the child begins to pull be t=0, and take the position of the wagon at that time to be x=0.

How fast will the wagon be going after 1.0 s of pulling?

Where will the wagon be then?

At what time will the wagon come to a stop (for an instant)?


What if

What if…?

  • What if I asked “where will the wagon be when it is going -1.0 m/s?”

  • You could:

    • find the time that v= -1.0 m/s

    • find the position at that time.


What if1

What if…?

  • Let’s find a generalization of that:

  • Where will object be when it’s velocity is v, given a known initial position, velocity, and constant acceleration?

simplify:


Derivatives and constant acceleration

Derivatives and Constant Acceleration

Yeah – consistency!


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