Chapter 3 - Exercises

1. Using Maple define the following functions and plot them simultaneously in a single graph

f1(x) = exp( - 0.3 x ² ) sin(x)

f2(x) = sin(x) cos(x)

2. Find accurately the intercepts of the functions f1(x) and f2(x) in the interval x={-p,+p}

3. Using Excel, define the functions

f(x) = sin(x) + cos(2x)

and

g(x) = sin(x)

Find the solutions of f(x) = g(x) within the interval x={-Pi..Pi}.

4. Solve question 3 using Maple.

5. Difficult problem (Maple)

The equations describing a projectile motion are:

x(t) = vox t                    (1)
y(t) = voy t + ( g/2 ) t2       (2)
vox = vo cos( q )
voy = vo sin( q )

where g = - 9.8 m/sec² is the acceleration of the projectile due to gravity, x(t) and y(t) the position of the projectile as a function of time, vox and voy the initial velocity components in the x and y directions, q the initial launch angle, and vo the initial speed of the projectile.

Find the launch angle, q, so as to hit a target located at X_T and Y_T.

Hints:

1. Define x(t) and y(t) as Maple functions; assign vox and voy
2. Solve for t in eq. 1
3. Replace in eq. 2 and make the resulting a function of 2 variables, t and q; call it Y(t,q)
4. Specify vo=36 m/sec, X_T=25m, and Y_T=17m.
5. Solve for q such as to hit the target
6. Plot the trajectory
7. There exists a second solution for q; find it
8. Plot the two trajectories simultaneously

6. Difficult problem (Maple)

Plot simultaneously the lemniscate function ( x² + y² ) ² = 4 ( x² - y² ) and the straight line y = 0.4 x + 0.25 and find the intersections between the curves.

Any questions or suggestions should be directed to Michel Vallières at vallieres@physics.drexel.edu