Examples Some lend themselves to computer solution!

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Presentation transcript:

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

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

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

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

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 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 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