ExamplesSome 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

• 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

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

• 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.

See Appendix H!

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