Output of prog. 1
B. With what speed does it land?
v0
45o
D
h
2. A golfer drives a ball with initial angle θ = 45o from a tee at the origin. The ball hits a window at height h = 10 m a distance D = 80 m away.
a. Find the initial launch speed v0of the ball.
b. Suppose a second golfer tees off from the same spot at an unknown angle and speed, and miraculously her shot also hits the window at the same height 1.8 s after launch. With what speed did it hit the window?
c. Was her ball ascending or descending or in level flight when it hit?
3. Suppose that a projectile is launched horizontally from a height of h = 10 m, at what speed must it be shot in order to barely clear (pass over) an obstacle of height 5.0 m placed on the ground 20 m away?
4. If a projectile is launched horizontally with initial speed v0 = 30ms, from a height h = 20 m, with what speed does it hit the ground? What is the angle of impact?
x y
v
0α
R
θ
5. A projectile is launched with initial speed v0 at angle θ with respect to the horizontal up a hill of angle α (slope is tan α). Compute the range of the projectile R (find how far from the launch point it lands on the hill.
6. In the previous problem, compute the launch angle that maximizes the range.
Ans. θ = π4+α2.
7. If a projectile is launched horizontally (in direction of increasing x) with initial speed v0, from a height h above the origin, find its speed s =
√
vx2+ vy2 as a function of x.
8. If a projectile is launched with initial speed v0, at angle θ from the origin, find its speed s =
√
vx2+ vy2 as a function of x.
9. If a projectile is launched with initial speed v0, at angle θ from the origin, find the angle its velocity vector makes with the horizontal as a function of x.
10. You can see that taking into account air resistance is pretty difficult if the motion is truly two or three-dimensional. If it is not, things may not be too complicated. Suppose that the effects of air resistance is to modify your acceleration by amount α|v|, always in a direction opposing your velocity. This is a good approximation, and α is usually small. Assuming α is small, a falling body will reach terminal velocity, it will reach a point where it falls at constant speed. Find this speed.
11. Suppose that the effects of air resistance is to modify your acceleration by amount α|v|, always in a direction opposing your velocity. Assuming α is small, find the velocity of a body as a function of time (let it be released at t = 0). If you get stuck, try using REDUCE.
12. Find the maximum height and range R attained by a projectile launched from the ground at v0 = 50ms, angle θ = 60o.
13. Find the speed and angle that v makes with the horizontal at x = R/4, R/2, 3R/4 by a projectile launched from the ground at v0= 50ms, angle θ = 60o.
14. A projectile is launched horizontally at v0, initial height h. Obtain a formula for its velocity vector as a function of time, and as a function of x, how far it has traveled from where it was launched.
15. A projectile is launched from ground level at v0, initial angle θ, from the origin. Obtain a formula for its velocity vector as a function of time, and as a function of x, how far it has traveled from where it was launched.
16. A golfer launches a ball from ground level at angle 45oto the horizontal, towards a wall 40 m away. After hitting the wall, the ball lands right on the tee from which it was launched! Find the speed with which it was launched.
17. A golfer launches a ball from ground level at angle 45oto the horizontal, towards a wall 40 m away. After hitting the wall, the ball lands 10 m from the wall. Find the speed with which it was launched. Note that when a projectile ricochets from a surface, the component of its velocity normal to (⊥ to) the surface gets reversed.
18. Find the maximum range and speed attained by a projectile launched horizontally at v0 = 50ms, initial height h = 20 m before hitting the ground.
19. Find the maximum height, speed and range attained by a projectile launched from height h = 20 m at v0= 50ms, angle θ = 60o before hitting the ground.
20. A golf ball launched from the origin at angle θ = 30omust clear a fence of height 20 m a distance 40 m from the tee. Find the minimal launch speed required.
21. A golf ball launched from the origin at initial speed v0= 80ms must clear a fence of height 20 m a distance 40 m from the tee. Find the minimal launch angle θ required.
22. Find the velocity as a function of time for a body falling from rest under the influence of gravity and air resistance such that ay=−g + b|vy|.
23. Use ode to find the maximum range of a projectile launched at 45o, v0 = 50ms with air resistance ay =
−g − 0.05 vy, ax=−g − 0.05 vx.
2.8. PROBLEMS 51
24. In the example with air resistance solved in Eq. 2.35, find the angle θ of launch that maximizes the range. The results will depend on b, g, v0. Hint; v0y= v0sin θ.
25. An ecological/biological model. Suppose that you have an isolated island with two species, bunny rabbits and wolves. Each year the bunny population grows in proportion to the number of bunnies, and shrinks because of wolf predation. The wolf population grows in proportion to the number of wolves, but shrinks if the ratio of wolves to rabbits gets to be too big. Use ode to study the populations as a function of initial populations, watch for crashes in populations. For example
# a is bunnies a’=0.06*a-0.05*b
# b is wolves b’=0.05*b-0.02*b/a
# those wolves are breeding nearly
# as fast as the bunnies, not good!
a=20
# try these wolf pops.
b=6
# b=10
# b=15
# b=20
# print wolf population print t,b
# print bunny pop.
# print t,a step 0,25
Note that if either population goes to zero, that species becomes extinct on the island, and you would need to modify the EOMS accordingly. Can you figure out how to make the bunny population explode if the wolf population goes down? We have assumed that only predation limits population, how could you model disease or overcrowding impact on the silly wabbits? Hint, try this, it produces a stable population
# a is bunnies a’=0.06*a-0.0025*a*a a=20
print t,a step 0,50
The new term describes detrimental effects of over-crowding. What is the asymptotic bunny population? Do you see that this is fundamentally the same equation of motion that we studied for air resistance?
26. Plotting trajectories with gnuplot is very easy; we simply tell gnuplot to graph x(t) and y(t) parametri-cally, such as in this example
set parametric x(t)=2*t y(t)=4.0-t**2 plot[0:1] x(t),y(t)
3 3.2 3.4 3.6 3.8 4
0 0.5 1 1.5 2
x(t), y(t)
Lets try it on the example with air resistance illustrated in Eq. 2.31
set parametric v0y=25
v0x=25 g=9.8 b=0.1
x(t)=v0x*t-0.5*b*v0x*t**2 y(t)=v0y*t-0.5*(g+b*v0y)*t**2
+0.1666*b*g*t**3 plot[0:4.6] x(t),y(t)
0 5 10 15 20 25 30
0 10 20 30 40 50 60 70 80 90
x(t), y(t)
Graph the case with v0= 40ms but with different launch angles (let b = 0.1) and by making graphs try to find the launch angle that gives the greatest range for this v0, g, b. Then graph the trajectory of the rocket Eq. 2.38 with v0x= v0y= 50, w = g/10, and find the range of the rocket by making graphs.
27. By graphing (with gnuplot) both trajectories Eq. 2.42, 2.43 with θ = 45o, ϕ = 10o, v0= 50 determine xmax, the place where the ball hits the hill.