average velocity

[slugger]

The accompanying figure shows the position versus time curve for a certain particle moving along a straight line. Estimate each of the following from the graph:
A) The average velocity over the interval 0 less than or equal to t less than or equal to 3
B) the values of t at which the instantaneous velocity is zero
C) The values of t at which the instantaneous velocity is either a maximum or a minimum
D) the instantaneous velocity when t=3s

the file attached contains the graph the x axis scales by ones and the y axis scales by 5s

 

[mmm4444bot]

Hello Slugger:

It looks like you tried to reproduce your graph using MS Paint software. Hopefully, your original is much larger and the curve is smooth.

When a curve represents object displacement versus time, then the slope of a tangent line is the instantaneous velocity at that point.

To answer questions about average velocity, you draw a secant line. Plot the two points on the graph where t equals zero and three. Connect them with a straight line, if you have multiple copies of your original graph with which to work. Estimate the corresponding s-coordinates, and then use the slope formula to determine the slope of this secant line. This slope represents the average velocity over the 3-second interval.

Any points where a tangent line slope is zero represent points where the instantaneous velocity is zero.

As you move along the curve from left to right (i.e., as time is elapsing), think about how a tangent line see-saws back and forth between positive and negative slopes. If you consider the slope of these many tangent lines in a particular vicinity of the graph, you will see that the slope attains a large positive or large negative value before it starts changing back toward zero.

Check out this animated illustration of a line tangent to a point moving along a curve.

Whenever the tangent line is steepest (either positively or negatively), then the slope at that point represents a maximum (when positive) or minimum (when negative) in that vicinity.

If you concentrate while watching the above animation, you can see a correlation between the velocity of the point and the slope of the tangent line which represents that velocity. When the tangent line is steepest, the point is moving the fastest.

All of the questions in this exercise can be answered by drawing lines and determining their slope from estimated values off the graph.

Cheers,

~ Mark

 

[slugger]

Mark~
Thank you, the explanation was awesome! I had thought of drawing a secant line, but because of a faulty book and an even faultier prof decided not too, this cleared things up completely! Thank you again so much!
~Scott