32 Practice Makes Perfect II

A mass hangs from a rope that goes through a pulley and is attached to a second mass that rests on a horizontal table (see picture below).  The hanging mass is a height from the floor.  The mass is also sufficiently large that it will start falling, dragging the mass along with it. The kinetic frictional coefficient between the small box and the table is .

Assuming that the table is long enough that the small mass comes to rest on the table, what is the total distance travelled by the small mass?

Hint: There are two mini-stories in this problem.  When does the first mini-story end, and the second mini-story start?

Now that we have our story straight, try solving the problem, starting with mini-story no. 1. If you get stuck, you can use the slider above to check whether your picture for story no. 1 matches mine.

Now story no. 2.  Try it yourself.  If you get stuck, you can use the slider below to check whether your picture for story no. 1 matches mine.

In an NFL (American) football match in 2021, the Ravens were down by 1 point with 3 seconds left in the fourth quarter.  Their kicker, Justin Tucker, would attempt a 66-yard field goal to win the game.  In football, the goal post is 10 ft above the ground.  Tucker’s kick would hit the post, and bounce in, winning the game for the Ravens.  Assume the ball is kicked at a 45-degree angle.

Assuming air resistance can be ignored (it can’t, but let’s pretend), determine how fast the football was moving immediately after the kick.

If you get stuck, go ahead and compare your drawing to mine, and then take it from there.

If you’re interested, Mark Rober has a fun video on his football-kicking machine.

One end of a spring is attached to a wall, and the other is attached to a toy car.  We let the x-axis be along the motion of the car, and set at the equilibrium position.  The car starts at rest.  Then, at , you give it a kick so that it has an initial velocity .   The car oscillates back and forth once every two seconds.

A. What is the angular frequency of the oscillation?

B. Write the appropriate expression for based on the amplitude and angular frequency.

C. Calculate .  Use your resulting expression to determine the amplitude of the oscillation.