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.
Part 1
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.
System: The boat and its contents will be treated as a point particle with 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
Application:
Immediate effects. 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.
A warning.
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.
Item for thought.
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?
Long-term effects.
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.
A warning.
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.
B.) Sketch rough graphs of your position and velocity as a function of time from the instant the fan was reversed.
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?
D.) Suppose that once you have reversed the fan, you keep the fan reversed and on. Will you arrive safely back at Base Camp?
Part 2
Suppose you are making your daily rounds. This involves:
- starting from rest at the Base Camp
- traveling to Station Two
- remaining at Station Two for the same amount of time it took to get there
- turning the boat around and traveling to Station One (without stopping at Base Camp)
- remaining at Station One for the same amount of time you were at Station Two
- turning the boat around and returning to Base Camp and stopping there
A.) Plot 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.
B.) Divide your graphs into segments and label each segment with the direction of the fan switch (forward or backward) during that segment.
Part 3 - Challenge
In Part 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?