Download Exercise 2

2.5. Unsolved Problems
Exercise 2.1: Exploding projectiles
A projectile is launched at 20 m/s at an angle of 30 with the horizontal. In the
course of its flight, it explodes, breaking into two fragments, one of which has twice
the mass of the other. The two fragments land simultaneously. The lighter one
lands 20 m from the launch point in the direction in which the projectile was fired.
Part (a) Where does the other fragment land?
Part (b) Plot the motion of the two fragments and the motion of the center of mass
point.
Part (c) Write a function that will plot the motion of the two fragments and the
motion of the center of mass when the initial velocity, mass of fragments, and position
of one of the fragments is given.
Exercise 2.2: Intersecting trajectories
A hunter with a blowgun wishes to shoot a monkey hanging from a branch. The
hunter aims right at the monkey and fires the dart. At the same instant, the monkey
lets go of the branch and drops to the ground.
Part (a) Show that the monkey will be hit regardless of the angle of the dart to the
monkey and the dart's initial velocity so long as it is great enough to travel the
horizontal distance to the tree before the dart hits the ground.
Part (b) Plot the points for the positions of the dart and the monkey when those
positions are separated by equal time intervals. Choose the time intervals so that the
last point is the intersection of the dart and monkey.
Part (c) Do the plot for several different initial values of the dart and for several
different heights of the monkey.
Exercise 2.3: Reflecting trajectories
A boy stands a distance x0 from a vertical wall and throws a ball. The ball leaves the
boy's hand at a height y0 above the ground with initial velocity v0. When the ball hits
the wall, its horizontal component of velocity is reversed and its vertical component
remains unchanged.
Part (a) Where does the ball hit the ground?
Part (b) Plot the motion of the ball for several different initial velocities.
Exercise 2.4: Principle of Galilean invariance
Two balls are released simultaneously from a platform above Earth's surface. One is
merely dropped from rest, while the other is launched with an initial horizontal
velocity v0. Graph the stroboscopic images of the two balls at successive instants of
time and show that they fall downward in unison, reaching equal heights at the same
time. Consider several different values of v0.
Exercise 2.5: Elastic collision
A ball moving with nonrelativistic velocity vi makes an off-center, perfectly elastic
collision with another ball of equal mass initially at rest. The incoming ball is
deflected at an angle  from its original direction of motion.
Part (a) Find the velocities and trajectories of the balls after the collision.
Part (b) Plot the trajectories and center of mass point for different values of vi and .
Exercise 2.6: Quadrupole
Consider three charges located on the z-axis.
One negative charge of magnitude q is
located at +s, the other, of the same magnitude, is at s. A positive charge of
magnitude 2q is located at the origin. The field point is at r, and it is assumed that
the separation s of the charges is small compared to the distance r.
Part (a)
Find the potential and electric fields to leading order in s/r.
Part (b) Plot the equipotential surfaces and the electric field lines.
Exercise 2.7: Fast Fourier transform
Consider the function
1
 1

Sin  2 t   1  Sin 6 t   Sin 8 t  
10
 5

Generate a table of data points from this function with random noise added.
Take
the Fourier transform of the table and plot the results.
Exercise 2.8: B field of a spinning disk
A thin disk of charge density s, radius R, and thickness t  R rotates with an angular
velocity  about the z-axis. Find the B field on the axis of the uniformly rotating
disk and plot the results.
Exercise 2.9: Fit to data
Consider the following data table:
In[l]:
data 
Table[{ x, 5 Sin[3 Pi x] + Random[ Real, {-.2, .2} ] } //
N //Chop[#,10^-3]&,
{x, 0, 0.5 ,0.05} ];
Part (a) Fit the data to a power series in x and plot the fit and data on the same
graph.
Part (b) Use leastSquares to fit the data to A Sin[B x] and plot the fit and data on
the same graph.