Intro Videos

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course PHY 231

PART 1 - 2Variables can be used to represent an event. The variables representing an event consists of at least two values that can be measured like speed and distance (A,B) . The average rate of change of event A with respect to B is equal to the change in A(distance) over the change in B(time). If A = 100 cm and B = 10 seconds than the average rate of change or in this case speed is 10cm per sec.

Example 2 Event 1: Car at position 1 at mile 15 at 8:00 am Event 2: Car at position 2 at mile 318 at 1:00 pm Find the Rate of change between event 1 and 2. Change A / Change B 318-15 / 5hrs (8:00 to 1:00) = 60.6 mph PART 3 Avg Velocity = rate of change in position with respect to time Avg acceleration = rate of change of velocity with respect to time Instantaneous velocity is the limiting value for average rates of change of time at a particular moment while time approaches zero. Instantaneous acceleration is the limiting value for average rates of change of velocity at a particular moment while time approaches zero. Part 4 Bar over a variable quantity means average. So that vBar = delta x/ delta t where x = position and t = time. Velocity at time t is equal to the change in x divided by the change in time. The limit as t approaches zero of x(t + delta t) - x(t) / change in t ->> which is the difference quotient. If x is a function of t then the instantaneous velocity is the derivative of v(x). Part 5 Graph of v vs. t : With two points (Vo, To) , (Vf, Tf) while Vf > Vo and Tf > To Delta T is the distance from To Tf as the change in time Delta V is the distance from Vo Vf as the change in velocity barA = delta V / delta T while delta t is = rise and delta v = run Therefore slope = barA If the line is straight the average velocity occurs at the midpoint of the interval So that the average velocity = Vf + Vo / 2 Part 6 Trapezoid rotated to match the previous graph where the bases are parallel to the y axis. The midpoint of the sloping segment of the trapezoid is equal to the line in the previous graph. The area underneath the line is the area of the trapezoid. The altitude from the midpoint of the line can be cut to form a right triangle that when rotated counterclockwise 90 degrees forms a rectangle that has the same area as the original trapezoid. Area = vBar delta T vBar delta x / delta t so delta x = vBar * delta t Part 7 V vs T graph You can use vertical lines to divide up a graph of intervals just like the trapezoidal rule from calculus that can be used to find the approximate area of a curve. For v vs. t, the velocity at any interval is approximate to the slope of the same interval. The integral from To to Tf of the function v(t)dt is equal to the change in position. The slope is equal to the approximate acceleration of that interval. Part 8 Same as part 7: Acceleration corresponds to the altitude per trapezoid. Average Acceleration = aBar * deltaT. The area represents the change in velocity in a given time interval. The sum of the trapezoids infinity is equal to the total change in velocity between To - Tf. " #$&*

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Intro Videos

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course PHY 231

Example 2

Event 1: Car at position 1 at mile 15 at 8:00 am

Event 2: Car at position 2 at mile 318 at 1:00 pm

Find the Rate of change between event 1 and 2.

Change A / Change B

318-15 / 5hrs (8:00 to 1:00)

= 60.6 mph

PART 3

Avg Velocity = rate of change in position with respect to time

Avg acceleration = rate of change of velocity with respect to time

Instantaneous velocity is the limiting value for average rates of change of time at a particular moment while time approaches zero.

Instantaneous acceleration is the limiting value for average rates of change of velocity at a particular moment while time approaches zero.

Part 4

Bar over a variable quantity means average. So that vBar = delta x/ delta t where x = position and t = time.

Velocity at time t is equal to the change in x divided by the change in time. The limit as t approaches zero of x(t + delta t) - x(t) / change in t ->> which is the difference quotient. If x is a function of t then the instantaneous velocity is the derivative of v(x).

Part 5

Graph of v vs. t :

With two points (Vo, To) , (Vf, Tf) while Vf > Vo and Tf > To

Delta T is the distance from To Tf as the change in time

Delta V is the distance from Vo Vf as the change in velocity

barA = delta V / delta T

while delta t is = rise and delta v = run

Therefore slope = barA

If the line is straight the average velocity occurs at the midpoint of the interval

So that the average velocity = Vf + Vo / 2

Part 6

Trapezoid rotated to match the previous graph where the bases are parallel to the y axis.

The midpoint of the sloping segment of the trapezoid is equal to the line in the previous graph. The area underneath the line is the area of the trapezoid. The altitude from the midpoint of the line can be cut to form a right triangle that when rotated counterclockwise 90 degrees forms a rectangle that has the same area as the original trapezoid.

Area = vBar delta T

vBar delta x / delta t so delta x = vBar * delta t

Part 7

V vs T graph

You can use vertical lines to divide up a graph of intervals just like the trapezoidal rule from calculus that can be used to find the approximate area of a curve. For v vs. t, the velocity at any interval is approximate to the slope of the same interval. The integral from To to Tf of the function v(t)dt is equal to the change in position. The slope is equal to the approximate acceleration of that interval.

Part 8

Same as part 7: Acceleration corresponds to the altitude per trapezoid. Average Acceleration = aBar * deltaT. The area represents the change in velocity in a given time interval. The sum of the trapezoids infinity is equal to the total change in velocity between To - Tf.

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