Differing Speed Scenarios

AP Calculus AB

February 2007

Acknowledgement

Mr Bob Dixon, Physics teacher extraordinaire, was essential to this project. He not only provided all the equipment; he also taught us how to use VideoPoint software.

Team 1Positive Acceleration, Positive

VelocityBy Emma

And Mrs. Clemens

Preparation

• First we obtained three meter sticks and set them out in a straight line to form our axis that the craft travels upon.

• We then used Mr. Dixon’s hovercraft as our vehicle moving along the axis.

• In my team’s demonstration, we started our craft at -3 and then floored the controls to the end of our meter sticks to demonstrate positive velocity and acceleration.

Video

Position

• The craft started at -3 meters, and as time went on it moved in the positive direction because of its positive velocity. It eventually reaches the origin.

• The graph is parabolic because velocity is increasing because of the craft’s positive acceleration.

Positive Velocity

• The craft started at rest, so velocity started at 0 m/sec.

• Then as time went on, velocity increased because of positive acceleration.

• If it was constant velocity, it would be a horizontal line, but the positive slope means acceleration factors in.

Positive Acceleration

• Acceleration is constant at around 0.742 m/sec2. (horizontal pink line).

• The acceleration actually fluctuates a bit, but overall it stays stays positive, and causes the craft to speed up as time goes on.

SpeedSpeed

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

time (sec)

m/s

ec

• The speed of the craft starts at 0 m/sec because the hovercraft was at rest.

• As velocity increases, speed increases until the craft is traveling at about 2.5 m/sec.

Summary

• Our hover craft started at rest at a position of -3 meters. It then positively accelerated with positive velocity up to and beyond the origin at 0.

• Velocity continued to increase as time went on, and acceleration fluctuated slightly between 1.175 m/sec2 and -0.076m/sec2, although its major trend is constant.

• Speed is the same as velocity, since the velocity was always positive.

Positive Velocity and Negative Acceleration

Team 2Levi and David

By Dave n’ Levi

Set Up

• First, we obtained a hovercraft. It modeled frictionless motion and allowed us to obtain results for position, velocity, and acceleration.

• We used meter sticks to scale our hovercraft’s motion.

Video

Using our data points on the graph of f, we see that f is increasing. However, we also see that the slope of f is decreasing. This means that although our hovercraft was moving forward, its velocity was decreasing, meaning that it was moving forward at a slower and slower rate.

(f(2)-f(1))/2-1 = .344 (f(21)-f(20))/21-20 = .092

This shows that for values early in the interval, the average velocity is lesser than the average velocity later in the interval. This shows negative acceleration.

Using our data points for the graph of f ', we see that the velocity of the hovercraft is positive, yet decreasing. This means that our hovercraft was getting slower while moving forward. The data table shows that the velocity values range from 4.124 m/sec to .8112 m/sec. Both of these values are positive, meaning that the hovercraft was moving forward, yet the hovercraft was clearly moving at a faster rate in the beginning. In this particular case the speed function is equivalent to the velocity function because the velocity in this example is always positive and speed is the absolute value of velocity.

Using the table values for f '', we see that every value of f '' is negative. The graph also shows this. Because f '' is always negative, this accounts for the hovercraft's "slowing down."

A better model for a(t) would be a constant function, a(t) = -1.38 according to the velocity function.

Summary of Data

• Our hovercraft experienced positive velocity and negative acceleration. This means that although our craft was always moving forward, it was slowing down. Our velocity was positive and decreasing, and our acceleration was (with some variations) a negative constant.

Visualizing Speed(Team 3)

By Ryan

&

Stacy

Set-Up

• We set up meter sticks to give a scale for the experiment.

• Used a hovercraft as the object undergoing the force changes to minimize friction.

• Videotaped the experiments to have a record.• As Team 3, we will be showing you negative

starting velocity combined with positive acceleration.

Negative Velocity + Positive Acceleration In Action

Position• As Team 3 we started our

hovercraft at 0.5 m on the “positive” side of the origin which was set at two meters to the right of the three meter sticks (basically we started a little to the left of the end)

• The hovercraft was heading in the negative direction because of its negative starting velocity.

• But went slower as time went on because acceleration was pulling it in a positive direction (was going backwards but was slowing down).

Velocity• Velocity is calculated by dividing the

distance traveled by the time it took to travel that distance.

• We started with a negative velocity which then became more negative because of human error (we should have clipped the video to start around 6 secs for only positive acceleration to be shown).

• Positive acceleration then began to slow the craft down

• The positive acceleration gradually made the velocity less and less negative as time went on.

• At the end you can see the velocity became slightly positive because of the continuation of the positive acceleration.

Acceleration

• As the derivative of velocity (a parabola), the graph is in a straight line.

• If we had clipped the video correctly, it would be horizontal at a positive y-value.

• When acceleration becomes positive the craft slows down.

Speed

• Speed is the absolute value of velocity so speed’s value is always positive as long as the object is moving.

• Our craft’s speed was highest in the beginning because the positive acceleration began slowing the craft down.

• Our speed would then begin increasing because the acceleration would begin to move the craft in a positive direction.

AP Calc Video Project

Group 4: Catherine and Maria

Video

The Data

The x-values are negative as time increases, with the y-values staying approximately the same – which makes sense, as the hovercraft in the video goes from right to left and not up or down. The x-values are becoming increasing farther apart, at an increasing rate, showing that the craft is becoming more negative in both velocity and acceleration.

The Graphs: Position vs Time

This graph shows position as a function of time. The hovercraft in the video begins around -1.5 m and moves away from the origin at an increasing rate, so it make sense that the graph shows the position as becoming more negative more quickly as time goes on. This parabola can be estimated by s(t)=-.44t2 + .0225t + -1.52, which judging from the r2 value of 1, is incredibly close to what happens on the video.

The Graphs: Velocity vs Time

This graph shows the velocity as a function of time, showing it starting at rest and becoming progressively more negative. In the video, the hovercraft starts at rest and become “faster,” but is going in a negative direction. Note that this, a line, is the derivative of position, a parabola. The function is estimated by v(t) = -.908t+.019, which is very close to s’(t) = -0.447t+0.0225.

The Graphs: Speed vs Time

This approximate graph shows the speed of the hovercraft as a function of time. Note that it is the absolute value of the next graph, velocity. In the video, the hovercraft starts at rest and gets increasingly faster, as is shown by this graph.

The Graphs: Acceleration vs Time

This graph shows acceleration as a function of time. Note that it, a generally constant line, is the derivative of the velocity graph, a line. In the video, the hovercraft accelerates at approximately the same rate, as shown by this graph. The acceleration is roughly -1 m/s2.

Summary

The hovercraft begins at rest, at x = -1.512 m, and then experiences negative velocity and negative acceleration as it “speeds up” in a negative direction over two seconds. It ends up at x = -3.268 m, with an almost constant y-value around y = 0.12 m. The acceleration is nearly constant as well, around -1 m/s2, with s(t)=-.44t2 + .0225t + -1.52, v(t)= -0.908t+.019, around -t, and speed as a function of time equal to |-0.908t+.019|. This can be seen in the table as well, as the values are becoming more negative and increasingly farther apart from adjacent points.

Overall SummarySpeed depends on the signs of both velocity and

acceleration: Going faster to the right: positive velocity and positive

acceleration Slowing down, but moving right: positive velocity and

negative acceleration Going faster to the left: negative velocity and negative

acceleration Slowing down but moving left: negative velocity and

positive acceleration

Conclusionobject is going faster when signs of velocity and

acceleration are the same and slowing down when signs of velocity and acceleration are opposite.