Exploring Forces
Name ___________________
Materials: Each group needs old motors and batteries, rulers, rectangular pieces of Styrofoam, access to a weigh
scale, and water to float the materials on. Use old motors and batteries to test balance because they may get wet
and you don’t want to ruin good ones. It helps to have some rectangular pieces of Styrofoam to illustrate what is
meant by question 4d.
1) If the boats don’t accelerate, this means that the
A) buoyant force is balanced by gravity
B) drag from the water is equal to the thrust
C) forward force is balanced by a backward force
D) thrust from the motors is constant
Explain: C) If the acceleration is zero, this means that the net force in all directions is zero (F = ma). In
particular, the forces with and against the direction of the velocity are balanced. Option A) is true, but is
referring to vertical forces and so it is not really relevant to a question talking about horizontal motion.
Option B) is true in terms of magnitude, but the forces are in opposite directions. Teacher should say that they
are equal and opposite. Option D) is probably true, but it doesn’t explain why there isn’t a constant
acceleration from this.
2) The propellers push the boat forward and drag works against this. The drag force is given by FD = ½ C d A v2.
What does each of the variables stand for? Explain how each variable applies to your boat and in other
everyday examples.
This equation will probably not be familiar to the students. You can have them make some reasonable
guesses and then teach the missing information. Drag may not be an explicit part of the course curriculum
but it is an example of another common force. The equation will only be used as a conceptual guide and it will
not be used in any mathematical problem solving.
v
This should be easy. It is the velocity (m/s) of the boat. Drag increases with speed. Walking in a pool
is easier than running. You feel less drag when riding a bicycle slowly compared to racing.
A
They should be able to guess that this is the area – but which area is not as clear. Have them consider
what it is like pushing a flutter board through water in different orientations. The relevant area is the crosssectional area perpendicular to the velocity.
d
They should be able to distinguish between the drag when you run in water vs. air. The difference is
due to the density of the fluid. Water is almost a thousand times denser than air.
C
So, what else affects drag? The shape of the object is very important; consider a speedboat vs. a ferry
boat or race car vs. a truck. This is the coefficient of drag. A hydrodynamic shape has a small coefficient of
drag.
3) The boat starts with a large acceleration but it quickly reaches a top speed, known as the terminal velocity.
Explain why the boat stops accelerating by referring to the drag equation and using force diagrams.
At the start, the velocity is zero and, therefore, the drag is zero.
As the boat goes faster, the force of drag increases.
When then drag becomes equal and opposite to the propelling force, it stops accelerating because F = ma;
however, it doesn’t stop moving. Forces tell you about accelerations, not about speed.
4) You want to reduce drag.
a) What will you do about d?
There is nothing that you can do. The race takes place in water.
b) What will you do about v?
You should NOT reduce drag by reducing speed – the competition is about having the highest speed!
c) What should you do about C? Use diagrams and words.
The shape should be streamlined in the direction of the velocity. It should be curved into a point at the front
and back. It should be long and narrow.
Top view
Side View
Note: The above analysis will be true for most boats. However, it is possible for a really well-made boat to
rise out of the water and skim over the top – reducing drag enormously. This requires a slightly different
shape and it is shown below. It needs a wide, gentle slope at the front.
d) What should you do about A? The cross-sectional area that is in the water determines how much
water must be pushed out of the way by the boat. The diagram below shows a rectangular boat that
floats with 2/3 of the body under the water. Shade the area that affects drag. Draw another 3-D
diagram of a rectangular boat that has the same volume as the first one but is twice as long and half
as wide. How does this change the cross-sectional area and, therefore, the drag?
You want to have A as small as possible – but how you do this is not obvious because it depends on the mass
of the boat and its dimensions. The long thin model will have HALF the cross-sectional area and, therefore,
HALF the drag. Therefore, you make A as small as possible by making the boat as long as possible. You also
make the mass of the boat as small as possible.
5) Conclusion: The length of your boat should be
A) as long as possible
B) as short as possible
C) a compromise
Explain:
A) If they have understood the previous question they will realize that it should be as long as possible – which
is the length behind the starting line of the trough, about 35 cm. Strangely enough, most students don’t get
this key idea and still think it should be a compromise. You need to decide if you should leave groups with this
misconception. If you don’t, you will have a great illustration of the physical principle, but the group will not
feel very successful. A longer boat also has other benefits. It will travel along a straighter path because it
can’t turn as much. This means it travels less distance and it also means that it hits the trough walls at a
smaller angle and will experience less friction. The only reason to make a boat any shorter would be to
provide some time to get up to top speed before hitting the starting line. Assure them that the boats hit
terminal velocity almost immediately.
6) Conclusion: The mass of your boat should be
A) as large as possible
B) as small as possible
C) a compromise
Explain:
B) The smaller the mass, the less volume that will be pushed into the water and, therefore, the less the
cross-sectional area and drag. This suggests that you should choose the lightest motor and propeller and
just one 9-V battery. However .....
C) .... the lightest motor might not spin as well as a heavier one and extra batteries may provide enough
extra thrust to more than compensate for the extra weight.
There is no clear right answer. Each group will need to make their own choices for the competition.
7) You should calculate roughly how much Styrofoam you need to float the motor and batteries. You can
assume that the mass of the Styrofoam itself is negligible. Hint: The density of water is 1g/cm3.
a) What is the mass of a 9-V battery?
Around 40 g
b) What is the mass of each motor type?
The masses will be similar to a 9-V battery.
c) What is the smallest volume of Styrofoam that can just float the heaviest motor and two 9-V batteries?
Give the groups some time to see if they can figure it out; then explain that the minimum volume of
Styrofoam displaces enough water to match the mass of the boat. The diagram on the left has more
Styrofoam than is needed, and the one on the right has just the minimum amount. This means that m = d V,
where the density is the density of water (1 g/cm3).
V = m/d = 120 g / 1 g/cm3 = 120 cm3.
d) You want to use more than this minimum amount of Styrofoam. Explain why.
You want to keep your batteries and motors dry. You want to be able to try more than one battery. You don’t
want to have to balance perfectly. The hull will shake and you don’t want to have a disaster where the front
dips under and then is pushed right under by the thrust. Each group needs to choose a safety margin of extra
Styrofoam.
e) Use your chosen length and the smallest volume of Styrofoam to determine the cross-sectional area of
your boat. Assume that the hull is a simple rectangle.
A = V/L = 120/35 ~ 4 cm2.
This is tiny!
8) Try floating some old batteries on Styrofoam. How wide should the boat be? Explain.
Balancing is hard because the mass is above the water surface. The boat needs to be as wide as possible –
just slightly less than the smallest width of the trough. Making the boat wider will also cause it to travel
straighter. However, the width of the trough is not uniform. The edges of the trough are somewhat wavy and
the calking that seals the middle sticks out a bit. The width needs to be just less than the smallest width of the
trough.
9) Use the calculated area from question 7 and draw - exactly to scale - what the front view of your boat hull
will look like.
It is important for students to draw this- otherwise, they won’t appreciate what the small area is telling them
and they won’t be prepared to build their hull.
This version won’t balance.
This one will be hard to slice and will break easily.
This is the best solution – a double hull.