Graphing 1D motion with constant acceleration.


Pictured here is the University of Manitoba Center for Earth Observation Science's air/ice boat "Skippy" (photo courtesy CEOS).  Skippy has a large fan at the back to allow it to accelerate as it slides across icy surfaces.  Suppose you are piloting a similar craft on very slippery ice which will not slow the boat at all if it is coasting.  Suppose also that the fan on your boat can be reversed instantanously, switching its direction of thrust from forward to backward.  Assume the action of the fan (when it is on) always produces an acceleration with the same constant magnitude.  Air resistance is negligible. 


This simple model of the air/ice boat is not realistic. We have chosen it to illustrate the properties of motion with constant acceleration. As you work through this example, decide which aspects are contradicted by your own experience. How would you develop a more realistic model of the boat's behavior? 




For the questions in this example, use the following coordinate system (illustrated above).  You have a Base Camp at position x=0.  Your assignment is to make observations of ice thickness and atmospheric conditions at two stations.  Station One is 1 mile east of Base Camp, and Station Two is 2 miles west of Base Camp.  Take east to be the positive x-direction. 









Problem 1








Part A



Suppose you have left Base Camp and are halfway to Station One.  You have been accelerating to the east the entire trip, but you now realize you have forgotten your gloves.  You immediately flip the fan control switch to backward, reversing the direction of the thrust.  Describe in words what will happen to your position and your velocity from the instant you reversed the fan. 



Solution



 System: 
The boat and its contents will be treated as a point particle.



 Interactions: 
An external force from the action of the fan (actually from the air that the fan is pushing).



 Model: 
One-Dimensional Motion with Constant Acceleration



 Approach: 






Once the fan has been reversed, the acceleration is in the opposite direction from the velocity. Thus, the boat will immediately begin to slow down. As it slows, it will continue to move east. In the language of our one-dimensional system, the velocity immediately begins to decrease with time, but the position continues to increase until the instant that the velocity has decreased to zero. 


It is very tempting to assume that as soon as the fan is reversed, the boat begins to move backward. This is not the case, however. Consider the case of a commercial jet landing at over 100 mph. The pilots will usually reverse the engines within a few seconds of landing to help slow the plane. If the plane were to immediately reverse direction, the effect on the passengers would be as if it had hit a brick wall and bounced off! Instead, the plane continues along the runway while gently decreasing its speed. 



One of the (many) reasons that your intuition might tell you that reversing the engine immediately reverses the direction is experience operating automobiles. In a car, the transmission is not designed to allow you to reverse the engine to slow the car. When the engine is in reverse either the car is moving backward or you are about to spend a few thousand dollars on your transmission. What feature do cars have (that is not present on the air/ice boat) to compensate for the fact that the engine cannot produce a significant acceleration in the direction opposite the car's motion? 


The motion is not finished when the boat stops, however.  If the fan is left in reverse, the continued thrust will now begin to accelerate the boat backward.  In one-dimensional language, the velocity reaches zero and continues to decrease, so the speed (absolute value of velocity) is now increasing.  The boat will therefore begin to move west, and with increasing speed.  Thus, the position vs. time graph will begin to decrease with a steepening slope.



It is vitally important to remember that an object cannot remain at rest for more than an instant if it is acted upon by a constant acceleration.









Part B



Sketch rough graphs of your velocity and position as a function of time from the instant the fan was reversed.



Solution



System, Interactions and Model: As in Part A.



 Approach: 






We construct the graphs that illustrate our answers to Part A.

  



For sketched graphs, you needn't put numbers on the axes, even though we did that for the position graph here.  It is important, however, that you make your time axes consistent with each other.  In this case, at the second division of the time axis, the velocity goes to zero.  Thus, on the position graph we expect to see the position's slope at zero (which it is).  Graders will always check for this kind of consistency.







Part C



  Given that the boat was halfway to Station One before you reversed it, that it started from rest at Base Camp, and that the fan was accelerating the boat forward until the instant you reversed it, where will the boat be when it (instantaneously) comes to rest after you have reversed the fan? 




System and Interactions: As in Part A.



 Model: 
To correctly answer this part, you will have to model the boat's motion starting from Base Camp.  The model is still One-Dimensional Motion With Constant Acceleration, but it has to be applied twice.  First, there is the eastward acceleration from the time that you leave Base Camp until you reverse the fan.  Then, there is the westward acceleration model of Part A. 



This technique of breaking up a complicated motion into more-easily-modeled pieces is both frequently used and extremely powerful.



 Approach: 






This time we are asked for a quantitative answer.  You can solve this with equations, but you would have to make up numbers. A better way to do this is to make use of the graphs as a tool to help you understand the symmetry of the equations.  Here is an extension  backward in time of the velocity graph of Part B which shows the acceleration from the forward thrust of the fan as you started out from Base Camp.  We know that the slope should simply be the opposite of the slope when the fan is reversed.  Further, we assume that you started at rest.  Thus, the red part of the graph is a reasonable representation of your motion with the fan accelerating you toward Station One.  Notice the obvious symmetry with the part of the graph from Part B that has been colored blue in the new graph.  It is clear that the area under these lines is the same.  Thus, using what we know about the relationship between velocity and position, we conclude that the distances traveled are the same.  Since the red part corresponds to traveling halfway to Station One, the blue part corresponds to the same distance.  The boat has made it to Station One just as it comes to rest (note that the blue part terminates at zero velocity).











Part D



Suppose that once you have reversed the fan, you keep the fan reversed and on.  Will you arrive safely back at Base Camp?




Solution



System, Interactions and Model: As in the previous parts.



 Approach:






You will certainly reach Base Camp, but it is perhaps not reasonable to say you have "arrived safely".  You might worry, since by the time you reach Base Camp you will be traveling fast enough that it would take you a mile to stop if you had to rely only on the reverse thrust of the fan.  Unless you can find some more efficient (but safe) way of stopping the boat, you will have to repeat the tedious process of turning the boat around again if you want to get your gloves. 














Problem 2



Suppose you are making your daily rounds.  This involves: 


    
	
  1. starting from rest at the Base Camp
    2. 
	
  2. traveling to Station Two
    3. 
	
  3. remaining at Station Two for the same amount of time it took to get there
    4. 
	
  4. turning the boat around and traveling to Station One (without stopping at Base Camp)
    5. 
	
  5. remaining at Station One for the same amount of time you were at Station Two
    6. 
	
  6. turning the boat around and returning to Base Camp and stopping there
    7. 










Part A



Sketch graphs of your position and velocity as a function of time for your entire daily rounds.  Assume that whenever you travel, you accelerate toward the destination for exactly half the trip and then decelerate for the other half.



System, Interactions and Model: As in the previous problem.



 Answer:








Note that the peaks of the velocity (solid dark red graph) do not occur at the same time as the position (dotted blue graph) peaks!  Rather, they correspond to changes in the curvature of the position graph.









Part B



Divide your graphs into segments and label each segment with the position of the fan switch (forward, backward or off) during that segment.  Remember that in your daily rounds, you always turn the boat before setting off from Base Camp or the Stations to point in the direction you are moving.




Solution



System, Interactions and Model: As in the previous problem.



 Approach:






In the plots shown here, green lines mean the switch is "forward", red lines mean "backward" and blue means "off".

 














Problem 3 - Challenge



In Problem 2 we assumed that the best way to travel is to speed up for half the trip and then slow down for the second half.  This gives your craft the greatest controllable speed (assuming you don't want to hit something at your destination to stop), but it also requires you to decelerate much earlier than if you just turned off the fan and coasted for a while at some intermediate speed.  Assuming you want to minimize the time to your destination, what is the best way to use the fan?