Marble Projectile Lab

Adam Seering

  In this lab, we rolled a marble off of a ramp and analyzed the path that it travelled.  We then analyzed the path that the marble took, and determined that it was a parabola.  We assumed that the parabola was of the form y=kx², and determined the value of k, using a hand-drawn best-fit line and a graphing utility.

  The purpose of this lab was to understand the motion of a projectile, and to be able to analyze it and predict it under varying circumstances.

  The first step of the lab was to set up a ramp for the marble to roll off of.  We used a ramp that rolled downhill at a significant angle at the start to give the marble a high velocity, but that leveled off at the bottom so that velocity would be only horizontal.  We then marked the ramp at a specific point so that we would always start the marble rolling at that exact point.

  We then taped graph paper to a piece of plywood parallel to and in front of the ramp, so that the marble would fall down in front of it.  We placed a metal barrier covered in carbon paper in front of the ramp, so that when the marble rolled off the edge of the ramp it would hit the paper in midair and make a mark at that point.  Using this tool, we measured the vertical position of the marble at regular horizontal intervals, and transferred the measurements to the graph paper.  Here is the resulting graph:

  I then entered the data from this graph into my graphing calculator and got the following screens:

  This was not an official part of the class lab, but I did it anyway because it was useful to determine that this was in fact a parabola, and to verify our later formulation of k.

  The next step was to calculate k.  To do so, we made a graph of y vs. x², where y was the vertical distance from the end of the ramp, and x was the horizontal distance.  The plot appeared to be roughly linear; because this was a graph of y vs x², y vs x would be parabolic.  This confirms the given formula, y = kx².  The slope of the line of best fit for the given coordinates was therefore equal to k.  We originally calculated k to be 0.028, a number that is supported by the results from the graphing calculator, and with later results from a computer-based graphing program, which generated the following graph:

  Here is the calculation that was used to determine k from the graph:

    y = kx²
    12 = k(21)²
    k = 0.0272

    16 = k(24)²
    k = 0.0278

    8.8 = k(18)²
    k = 0.0271

    The average of 0.0271, 0.0272, and 0.0278 is k = 0.0274.

  The next step was to use our values for k to predict where the marble would land in a specific situation.  The goal was to have it launch off of the ramp with the ramp on top of a table, and to have the marble land in a cup on the floor.  We measured the distance from the bottom of the ramp to the top of the cup, and plugged that value into our equation as y.  We solved for x, and positioned the cup accordingly.  The ball landed at the predicted distance, and it would have landed in the cup had the cup not been slightly to one side of the marble's plane of travel.  We tried again after a quick realignment, and the marble went right into the cup, showing that the equation and our value for k were, indeed, correct.

  The marble then proceeded to bounce right out of the cup, showing conclusively that the equation did not take into account the floor.  Analyzing this effect will likely be the topic of another lab.

  This lab had a relatively small potential for error.  This makes sense, as the graph produced by the data gathered in the lab aligned closely with expectations, and predictions made using the data turned out to be essentially correct.  The greatest potential source of error in the lab was in not taking into account factors that could affect the transit of the ball.  The ramp might not have been flat at its end, meaning that the ball would start out with a vertical component to its velocity, something that we assumed not to be the case.  Air resistance, as has been true in all labs so far, was also ignored; this was probably fairly safe in this case, however, as we were working with a small, dense, round object travelling at relatively low speeds; each of these factors indicates a minimal effect of air resistance.  Other than that, there really weren't any other factors that could perceptibly influence the results in this lab.

Questions:

1) x = v_xt, so x² = v_x²t²
  Divide y=(1/2)gt² through by this, and you get:
  y/x² = (1/2)(g/v_x²)
  or y = g/(2v_x²)(x²)

  if k = g/(2v_x²), substitution gives us:
  y = kx²

2) Launching the ball from a lower point on the ramp would decrease the horizontal veloty of the ball; x would decrease, so k would have to increase for any given vertical point y.

3) a) height = 1.143 meters
    V_i = 0
    a = g
    d = V_it + (1/2)at²
    1.143 = 0 + 4.9t²
    t = 0.483 sec

  b) d = vt
    0.663 m = v(0.483)
    v = 1.373 m/sec

  c) V_f² = V_i² + 2ad
    V² = 0 + 2(9.8)(1.143)
    V = 4.73 m/sec

  d) V_total² = V_x² + V_y²
    V² = 24.288
    V = 4.93 m/sec²

4) d = V_it + (1/2)at²
    3658 m = 0 + 4.9t²
    t = 27.3 sec

  This lab demonstrated effectively the parabolic path of a projectile, and how one would go about predicting that path.  It also showed that such predictions can be quite accurate without putting in a great deal of effort towards increasing precision.  This was because, as I explained earlier, there were fairly few causes for error present in the lab.  It effectively completed its purpose: to demonstrate how to analyze projectiles.