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Calculating Velocity
Problem 7.59*
Riding a bicycle into headwind
By: Andrew Quinn
Introduction and given variables
Joe can pedal his bike at a velocity of 10 m/s
when there is no wind.
His bike has a rolling resistance of 0.80
N*s/m.
Joe and his bike’s drag area is CdA =
0.422m^2.
We will assume that the density of air is 1.2
kg/m^3.
The mass of Joe and the bike is constant in
this problem; therefore it is negligible.
Problem
The purpose of the
problem is to:
1) develop an equation for
the speed at which Joe
can pedal into a
headwind of 5 m/s.
2) solve for the velocity.
3) explain why his velocity
into the wind is not 5
m/s.
Step 1: Calculate his force.
First of all, we must
calculate the
amount of force
that Joe is exerting
when there is no
wind
That force is equal
to the frictional drag
of him and his bike
while riding in
addition to the
friction due to the
rolling resistance of
his bike.
Calculations
We can calculate the drag of Joe and
his bike by using the following equation
Eq. 7.62 CD = drag/ (1/2*r*V^2*A)
Rearranging this equation, we solve for
the drag.
Drag = CDA*1/2*r*V^2
Substituting the given values, we find
that the drag = 25.32 Newtons
Calculations (continued)
Next, we calculate the friction due to the
rolling resistance of the bike.
This is done by multiplying the rolling
resistance by the velocity.
Friction due to rolling resistance = 0.8 N*s/m
* 10 m/s = 8 Newtons
This value is added to the drag calculated
earlier (25.32 Newtons) to determine the
total force he is overcoming which equals
33.32 Newtons.
Step 2: Calculate his power.
We now can calculate Joe’s power by
multiplying this force (33.32 N) by his
velocity of 10m/s.
Power = 33.32 N* 10 m/s = 333.2
watts (0.45 hp)
This is Joe’s power in any situation he is
riding in.
Solving for Velocity
We must now solve for V when there is a
headwind of 5 m/s.
Joe’s power equals the sum of the resistive
forces multiplied by the velocity.
333.2 W =
V*[CdA*r/2(V+5m/s)^2+0.8kg/s*(V+5m/s)]
V^3 + 13.16 m/s * V^2 + 25 m^2/s^2 * V –
1300.63 m^3/s^3 = 0
V = 7.37 m/s
Discussion
This problem helps us visualize how
wind resistance works.
Rolling resistance of an object is linearly
related to the headwind.
On the other hand, air resistance is
quadratically related to the headwind.
This is why a headwind of 5 m/s does
not slow someone down by 5 m/s.
Biomedical Application
It is hard to relate this problem to a
biomedical application inside the body.
However with this information, we can see
how one can calculate the endurance of a
biker by knowing the current winds that he is
riding in.
A biker’s energy level is an important factor to
consider when monitoring his health.