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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

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

• 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

• 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

• 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

• 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

• 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

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

• then its velocity changes at a constant rate:

• And its position changes quadratically with time:

Position with Constant Acceleration

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

### 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 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 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

Yeah – consistency!