Download Sect. 2.4, Part II

Examples
Some lend themselves to computer solution!
• Example 2.4: Simplest example of motion with a
retarding force: Find the velocity v(t) & the
displacement x(t) of 1d horizontal motion in of a
particle in a medium in with retarding force
proportional to the velocity. Fr(v) = - mkv.
Initial conditions: at t = 0, x = 0, v = vo
 x = 0 , v = vo
Worked on the board!
• Example 2.5: Find the
velocity v(t) & the
displacement z(t) of a
particle undergoing 1d
vertical (falling) motion
in Earth’s gravity, if the
retarding force is
proportional to the
velocity. Fr(v) = - mkv.
Initial conditions:
at t = 0, z = h, v = vo
 z = h , v = vo
Worked on the board!
Example 2.5: Numerical results for “free fall”
velocity versus time with air resistance
• Example 2.6: (A Physics I Problem!) Projectile motion
in 2d, with no air resistance. The initial muzzle
velocity of projectile is vo & the initial angle of
elevation is θ. Find the velocity, displacement, &
range. Initial conditions: at t = 0, v = vo, x = y = 0

x = y = 0 , v = vo
 vxo = vo cosθ, vyo = vo sinθ
• Example 2.7: (Nontrivial!) Projectile motion in 2d, with
air resistance. Initial muzzle velocity = vo, initial
angle of elevation = θ. Retarding force proportional
to velocity: Fr(v) = - mkv. Find v(t), x(t), y(t), &
range. Initial conditions: at t = 0, v = vo, x = y = 0
 x = y = 0 , v = vo
 vxo = vo cosθ  U
vyo = vo sinθ  V
Example 2.7: Numerical results for trajectories
for various values of retarding force constant k
Example 2.7: Numerical results for the range
for various values of retarding force constant k
See Appendix H!
• Example 2.8: Use the data shown in Fig. 2-3
to (numerically) calculate the trajectory for an
actual projectile. Assume:
vo= 600 m/s, θ = 45°, m = 30 kg. Plot the
height y vs the horizontal distance x &
plot y, x, & y vs. time both with & without air
resistance. Include only air resistance &
gravity. Ignore other possible forces such as lift.
Example 2.8: Numerical results
Figure for Problem 3, Chapter 2
• Example 2.9: (A Physics I Problem!) An Atwood’s machine
= smooth pulley & 2 masses suspended from a massless string
at each end. Find the acceleration of the masses & the tension
in the string when a) the pulley is at rest & b) when the pulley
is descending in an elevator at a constant acceleration α.
• Example 2.10: A charged particle moving in a
uniform magnetic field B. Find motion of particle.
Initial conditions: at t = 0, x = xo, y = yo, z = zo,
vx = xo vy = yo, vz = zo