M-5  Projectile Motion

OBJECTIVE: To find the initial velocity and predict the range of a projectile.
APPARATUS:

Ballistic pendulum with spring gun and plumb bob, projectile, single pan balance, elevation stand.

PART I. BALLISTIC PENDULUM

Figure 1: The spring gun.
Figure 2: A side view of the catcher

INTRODUCTION:

A properly aligned spring gun shoots a ball of mass $m$ into a pendulum catcher of mass $M$. See Figs. 1 and 2. The pendulum traps the ball; thereafter the two move together in pure translation. Since ball and pendulum have no relative motion, the collision is inelastic and thus does not conserve mechanical energy (where does it go?). Of course linear momentum is still conserved, and hence momentum of ball before impact, $\mu$, equals the momentum of ball plus pendulum after impact, $(m+M)V$:

$$mu=(m+M)V$$ (1)

where $u$ = ball's velocity before impact and $V$ = initial velocity of combined pendulum plus ball.

To find $V$, note that although the impact doesn't conserve mechanical energy, the motion after impact is almost frictionless and thus conserves mechanical energy. Hence the kinetic energy of the ball plus pendulum at $A$ in Fig. 2, just after impact, equals the potential energy of the two at the top of the swing (at $B$). Thus

$$\frac{1}{2}(M+m)V^{2}=(M+m)gh .     \mbox{ Hence, }     V=\sqrt{2gh} .$$ (2)

Pre-lab Quiz

You should be able to complete this brief quiz before proceding.
Read the Introduction section carefully, especially if you are unfamiliar with the concepts of conservation of energy and momentum.

• Expt. M05 Quiz

ALIGNMENT:

If properly aligned, our bifilar type of suspension for the pendulum (see Figs. 1 and 2) prevents rotation of the bob of mass $M$. Hence the motion is pure translation. To ensure proper alignment, adjust the three knurled screws on the base so that

A.

A plumb bob hangs parallel to the vertical axis of the protractor, and

B.

The uncocked gun axis points along the axis of the cylindrical bob. (You may need to adjust the lengths of the supporting strings.)

SUGGESTED PROCEDURE: 

1. Find $m$ and $M$ by weighing on the single pan balance.
2. Find the height $h$ by measuring angle $\theta$ and the length $L$ of the pendulum.
   NOTE: $h=L-L\cos\theta$ and $L$ is not the length of the string, (see Fig. 2).
   Suggestions on finding $\theta$: Find an approximate $\theta$, and then make a masking tape slit on the back of the protractor at this approximate $\theta$ so your eye can locate the proper viewing area to find a more accurate $\theta$ on subsequent firings. To avoid parallax (see Appendix 4) in reading the protractor choose a line of sight determined by the string in front of the protractor and the string in back of the protractor. Careful observation after a little practice will enable you to get $\theta$ to within a degree.
3. Calculate the initial velocity $V$ of the combined pendulum bob and ball.
4. Calculate the initial velocity of the ball, $u$, as it leaves the gun.
5. Estimate the uncertainty in $u$. [Hint: Since the largest uncertainty is likely $\Delta V$, then $\Delta h$ is important. While $h$ is a function of the measured $L$ and $\theta$, the uncertainty in the angle measurement, $\Delta \theta$, will probably dominate. To manually calculate the uncertainty in the resulting trigonometric function, sometimes it is simpler to calculate separately the function for $\theta+\Delta \theta$ and for $\theta-\Delta \theta$. Otherwise the smart form can do this for you. Remember to use absolute, not relative errors when propagating errors through addition error propagation

PART II. RANGE MEASUREMENTS HORIZONTAL SHOT:

1. After finding $u$, (the velocity of the ball leaving the gun) predict the impact point on the floor for the ball when shot horizontally from a position on the table.
2. To check your prediction experimentally:
   

   (i)

   Use the plumb bob to check that the initial velocity is horizontal.

   (ii)

   Measure all distances from where the ball starts free fall (not from the cocked position). All measurements of course refer to the bottom of the ball so $x = 0$ corresponds to the radius of the ball beyond the end of the gun rod. Check that the gun's recoil does not change $x$.

   (iii)

   Fasten (with masking tape) a piece of computer paper at the calculated point of impact, and just beyond the paper place a box to catch the ball on the first bounce.

   (iv)

   Record results of several shots. (The ball's impact on the paper leaves a visible mark.) Estimate the uncertainty in the observed range.

   (v)

   Is the observed range (including uncertainty) within that predicted.

3. Work backwards from the observed range to calculate the initial velocity $u$. This $u$ is probably more accurate than the value obtained with the ballistic pendulum.

ELEVATED SHOT:

1. Use the stand provided to elevate the gun at an angle above the horizontal. The plumb bob will give the angle of elevation, e.g. 90$^o$ - protractor reading. For the elevated gun, be sure to include the additional initial height above the floor of the uncocked ball.

2. Before actually trying a shot at an angle, again predict the range but use the value of $u$ which you found from the horizontal shot. (See item 3 above).

3. Make several shots, record the results and compare with predictions.

QUESTION:

From the measured values of $u$ and $V$ in Part I of this experiment, calculate the kinetic energy of the ball before impact, $\frac{1}{2}mu^{2}$ and the ball and pendulum together after impact, $\frac{1}{2}(m+M)V^{2}$. What became of the difference?

OPTIONAL:

$$\mbox{1.  Show that for momentum to be conserved:   } \frac{K... ..._{\small\mbox{after impact}}}=\frac{m+M}{m} \mbox{            }$$

2. Find the spring constant $k$ of the gun from $\frac{1}{2}mu^{2}=\frac{1}{2}kx^{2}$.