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Predicting The Range Of A Projectile

You will be given a bow and arrow and asked to determine the distance the arrow will travel at some arbitrary draw of the bow and angle of inclination. Once you determine this value, it will be checked by an actual firing of the arrow. Good luck!

In drawing back the bow from its equilibrium position, work is being done against it and effectively, potential energy is stored when the string is pulled back. When released, the energy is transformed into kinetic energy. Since kinetic energy is written as

K.E. = 1/2mv2 (1)

The velocity can then readily be determined as:

v = 2 KE / m (2)

With the initial velocity known, the angle of inclination can be set and the range determined from the equations of motion:

R = vo2 sin(20) / g (3)

The problem then, is to determine how much energy is stored in the bow when drawn back a predetermined amount. Work is merely the product of:

E = W = F s (4)

If one plots the force necessary at each distance (s) that the string is drawn back, then the area under the curve is the total work done or the total energy stored in the system, available to be transformed into kinetic energy. You can integrate this function to determine the area under the curve, but since the draw curve is not linear, you do not know the function of force and distance. Instead, you will add up the values of simple geometric shapes to determine this area.

OBJECTIVES:

  1. To predict the range a given projectile will travel given the angle of inclination and the opportunity to determine the initial velocity.
  2. To numerically determine the area under a curve and understand a physical meaning of area under a curve as so often determined through techniques of integration.

APPARATUS:

PROCEDURE:

  1. Determine the force necessary to pull the string back various amounts up to full draw. This should probably be determined for each centimetre from equilibrium to full draw.
  2. Plot the values on linear graph paper.
  3. Numerically determine the area under the curve. This may be done by adding up narrow trapezoids of width 1 cm and a measured height. Perhaps you can approximate the curve for Force as a function of distance with a known function. Then you can perform normal integration. In the limit as the trapezoids become infinitely narrow and the number of them becomes infinite, adding them up is the same as integration. When an algebraic function relating the variables is not known, the results must be determined numerically. Hint! Use the Trapezoidal Rule (remember from Calculus I?).
  4. Measure the mass of the arrow.
  5. From the area under the curve we know the kinetic energy. Assuming 100% of this energy is transferred (perhaps some is lost in the string grazing your arm in firing, the "twang" of the string, etc), determine the initial velocity and range of the projectile.
  6. Test your prediction and calculate % error.
  7. Your write-up must include a complete estimate of uncertainties. Remember that one value that is small compared to another more dominant one can justifiably be neglected.
  8. In firing the arrows, exercise CAUTION and follow SAFETY rules described by your instructor.
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