2. 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
Worked on the board!
 x = 0 , v = vo

3. vertical (falling) motion in Earth’s gravity, if the
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 Worked on the board!
 z = h , v = vo

4. Example 2.5: Numerical results for “free fall”
velocity versus time with air resistance

5. Example 2. 6: (A Physics I Problem
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

6. 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

7. Example 2.7: Numerical results for trajectories for various values of retarding force constant k

8. Example 2.7: Numerical results for the range for various values of retarding force constant k

9. 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.

10. Example 2.8: Numerical results

11. Figure for Problem 3, Chapter 2

12. Example 2. 9: (A Physics I Problem
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.

13. Example 2. 10: A charged particle moving in a uniform magnetic field B
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