PHYS 105: Introduction to Computational Physics
Spring 2009
Homework #5
(Due: May 21, 2009)
1. Do in-class exercise 7.1. Turn in your program (part 2) and three plots (parts 4 and 5), and
clearly state the results you obtain in response to the other questions asked.
2. Consider again a projectile moving in two dimensions under the combined effects of gravity
and air resistance. Initially the projectile is launched from x= 0, y = 0 with speed v0= 100
m/s at an angle of θ0= 60âŚto the horizontal. The components of its acceleration are
ax=âβ|v|vx,
ay=âgâβ|v|vy,
where β= 0.001 (chosen so that the initial acceleration for this value of v0is the same as the
case Îą= 0.1 in problem 1).
(a) Compute the projectileâs range and time of flight (take δt = 0.01, and donât forget to
interpolate!), and compare them to the results of the ι= 0.1, β = 0 calculation in
problem 1. Plot both trajectories (ι= 0.1, β = 0 and ι= 0, β = 0.001) on the same
graph.
(b) At what angle to the horizontal θ1does the projectile hit the ground in either case in
part (a)? What would θ1be in the absence of air resistance (i.e. ι=β= 0)?
(c) By what factor (to within 1 percent) must the launch speed for ι= 0, β = 0.001 be
increased to restore the range to the ι=β= 0 result?
(d) By varying the value of θ0, determine the maximum range of the projectile for v0= 100
m/s (again with ι= 0, β = 0.001). To what value of θ0(to 1 decimal place) does this
correspond?
(e) For v0= 100 m/s, plot θ1as a function of θ0.
3. Now suppose that the value of βin problem 2 varies with height y, according to the law
β(y) = 0.001eây/h ,
where h= 500 m (not a very realistic description of Earthâs atmosphere!). How does the
maximum range of the projectile (as computed in problem 2d) change as a result? What if h
= 5 km (a much better approximation to reality)?