micha ran 2 miles north in 20 minutes. she then turned back and ran 1 mile in 15 minutes. to cool down, she walked the rest of the way home. it took her 20 minutes.

what was micha's average velocity for her run north?

what was micha's average velocity for the whole trip?

what was her average speed for her entire jog?

Explanation:

Given data,

The distance Micha run towards the north, d₁ = 2 miles

The time taken by Micha to run 2 miles is, t₁ = 20 min

                                                                     = 0.33 h

The displacement towards the south, d₂ = -1 mile

The running duration, t₂ = 15 min

                                        = 0.25 h

Walked displacement, d₃ = -1 mile

The time period to cover that distance, t₃ = 0.33 h

1) The average velocity towards north,

                                  v₁ = d₁ / t₁

                                      = 2 / 0.33

                                      = 6.06 miles/hr

Hence, Micha's average velocity for her run north, v₁ = 6.06 miles/hr

2) The average velocity for whole trip,

                         V = (v₁ + v₂ + v₃) / 3

     The velocity towards south,

                         v₂ = d₂ / t₂

                              = -1 / 0.25 h

                              = -4 miles / hr

  The walked velocity,

                           v₃ = d₃ / t₃

                               = -1 / 0.33

                               = - 3.03 miles/hr

∴ The average velocity,

                        V = (v₁ + v₂ + v₃) / 3

                             = (6.06 - 4 -3.03)/3

                             = - 0.32 miles/hr

Hence, Micha's average velocity for the whole trip, V = - 0.32 miles/hr

3) The average speed for her entire jog,

      The velocity of her jog, v₃ = - 3.03 miles/hr

∴ Speed,

                     s = |v|

                        = 3.03 miles / hr

Hence, the speed of her jog is, s = 3.03 miles / hr                

Explanation:

Notice that the pendulum's mass is 0.1 kg, the length of the string is 9.4 m, the starting angle is 0.1 radians and it is started without imparting any velocity (just letting the mass drop). therefore, the differential equation for the pendulum's motion can be obtained by decomposing the forces acting on the pendulum into

those along the string (tension and component of the pendulum's weight along the string's direction) and the perpendicular to such (tangential to the pendulum's motion. Notice that the Net Force in the system is that associated with the tangential component of the weight (see attached image)

For small angles (which is our case for the angle of 0.1 radians),  can be approximated by the first term of this function’s Taylor expansion: , thus leading to the equation:

which dividing both sides by the pendulum's mass, and  replacing the tangential velocity with the angular velocity times radius of motion (length of pendulum), becomes:

And for our given values, this becomes: