Use units to perform physics calculations in both SI and Imperial units. Compute with units the terminal velocity of a falling paratrooper by modeling the deceleration of velocity due to drag.
Imagine that a paratrooper is dropped from an airplane. Assume there are only two forces acting on the paratrooper, the gravitational force and an opposing drag force from the parachute.
The net force acting on the paratrooper can be expressed as :
is the paratrooper's mass
g is the acceleration due to gravity
is the paratrooper's velocity
is the drag constant
Define and Solve the Differential Equation
Define the differential equation describing the balance of forces.
syms g m c_d syms v(t) eq = m*diff(v(t),t) + m*g == c_d*v(t)^2
Assume that the parachute opens immediately at so that the equation
eq is valid for all values of . Solve the differential equation analytically using
dsolve with the initial condition .
velocity = simplify(dsolve(eq, v(0) == 0))
Find the units of the drag constant in SI units. The units of Force are Newton or expressed in SI base units are . Since they are equivalent they have a unit conversion factor of 1.
u = symunit; unitConversionFactor(u.N, u.kg*u.m/u.s^2)
Since the drag force must have the same physical dimension in Newton as the gravitational force , the physical dimension of can be solved for.
syms drag_units_SI drag_units_SI = simplify(solve(drag_units_SI * (u.m / u.s)^2 == u.N))
Acceleration due to gravity
Substitute these values into the velocity equation and simplify the result.
vel_SI = subs(velocity,[g,m,c_d],[9.81*u.m/u.s^2, 70*u.kg, 40*drag_units_SI])
vel_SI = simplify(vel_SI)
Compute a numerical approximation to 3 significant digits.
digits(3) vel_SI = vpa(vel_SI)
The paratrooper soon approaches a constant velocity when the gravitational force is balanced by the drag force. This is called terminal velocity and occurs when the drag force from the parachute is roughly equivalent to the gravitational force, and there is no further acceleration. Find the terminal velocity by using
limit as .
vel_term_SI = limit(vel_SI, t, Inf)
Rewrite Velocity Using Imperial units
Finally, we rewrite the velocity function from SI units to Imperial units.
vel_Imperial = rewrite(vel_SI,u.ft)
Rewrite the terminal velocity.
vel_term_Imperial = rewrite(vel_term_SI,u.ft)
To plot deceleration, we measure the time t in seconds and replace t by t = T s, where T is a dimensionless symbolic variable.
syms T vel_SI = subs(vel_SI, t, T*u.s)
vel_Imperial = rewrite(vel_SI, u.ft)
Separate the expression from the units by using
separateUnits. Plot the expression using
fplot. Convert the units to strings for use as plot labels using
[data_SI, units_SI] = separateUnits(vel_SI); [data_Imperial, units_Imperial] = separateUnits(vel_Imperial);
We see that the paratrooper's velocity approaches its steady state when . Show how the velocity approaches terminal velocity by plotting the velocity over the range .
subplot(1,2,1) fplot(data_SI,[0 2]) title('Deceleration in SI Units') xlabel('Time in s') ylabel(['Velocity in ' symunit2str(units_SI)]) subplot(1,2,2) fplot(data_Imperial,[0 2]) title('Deceleration in Imperial Units') xlabel('Time in s') ylabel(['Velocity in ' symunit2str(units_Imperial)])
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