Linear Acceleration and Projectile Path

Transcription

1 Linear Acceleration and Projectile Path Objective To study the motion of a body when projected at an angle above the horizontal on an inclined plane. Introduction In this experiment we ll use an air table to study projectile motion. The air table allows one or two stainless steel pucks to glide on a glass surface using a cushion of air, similar to an air-hockey game. If we place a piece of paper underneath the puck, we can record its trajectory by use of a spark apparatus, which leaves a trail of dots on the paper. More on this later. Theory You may recall the formula for an object s motion with constant acceleration in one dimension (x): x = x + v t at2, where x is the object s position at any time t (instantaneous position), x is the displacement at time t = 0 (initial displacement), v is the speed at time t = 0 (initial speed) and a is the object s acceleration. The mathematics for considering motion in a second independent dimension y follows exactly, but we need to be careful - we have separate accelerations, displacements and velocities in both x and y, as shown in the general equations below: x = x + v x t a xt 2, (1) y = y + v y t a yt 2 (2) Notice how we ve distinguished between the initial positions, displacements and accelerations of x and y components by the subscripts. These are important, so be sure to keep track of them so that you know in which direction you re going!

2 2 y v 0 α v y0 y 0 v x0 x 0 x FIG. 1: Top and side view of the air table used in the experiment. How do x and y relate to each other? The diagram in figure 1 should help clear things up. The quantity v is the total initial speed, which is related to v x and v y through the launch angle α, by v x = v cos α and v y = v sin α (3) To simplify things, we re going to assume that x = y = 0 for this experiment. This is ok, because we re always allowed to pick the origin of our co-ordinate system (once). The x and y directions will be given by the plane of our air table. Normally, the air table sits flat and level. You might well realize that in this orientation, there shouldn t be any acceleration in either the x or y directions. However, if we tilt the air table (say along the y axis), we can produce a gravitational acceleration in the y direction only. This means that a x = 0, and a y (which depends on the tilt of the table) can be written as a y = a, where we introduce the - ve sign to indicate that the acceleration is in the - ve y direction. We can summarize the above by rewriting equations (1) and (2) as x = (v cos α) t (4) y = (v sin α) t 1 2 at2 (5)

3 3 In this experiment you ll measure the range (R) and maximum y displacement (y max ) of your puck to see how well the equations of motion above are able to predict the trajectory. The range is the horizontal distance travelled after y has returned to zero, and is given by and the maximum y displacement is given by R = v 2 sin 2α, (6) a y max = v 2 sin 2 α. (7) 2a You can look at figure 2 for a schematic representation of these quantities. Apparatus We make use of a standard air table complete with a puck, a clear plastic launching ramp and a spark timer for indicating equal time increments dt while the puck is in motion. The spark timer fires at a fixed rate, leaving small burn marks on the underside of the paper placed between the puck and glass table. The spacing of these dots is thus directly proportional to the puck s speed. Procedure Prepare the air table with a clean sheet of paper, checking first that the carbon paper is free of scratches or tears. Make sure that the paper is flush with the bottom of the table and that it is smooth. There are actually two pucks on the air table. The extra puck should be immobilized on the table, using the coral provided, and its tubing kept out of the way. It s very important that you keep this second puck on the air table, as the spark timer won t function without it. Position the launching ramp in a bottom corner of the table with the side marked AG horizontal so that the side marked CD is inclined to the horizontal, as in figure 2. Mark the two lines CD and DE so that the lines can be transferred to the other side of the paper. The value of the launch angle α is approximately 50, but you should measure it yourself to be certain. Place the piece of sponge rubber along the bottom edge of the table opposite

4 4 y max D E R B C α F A G FIG. 2: to the launching ramp, to absorb the shock of the landing puck. With the air on, make one or two trial launches of the test puck from the ramp CD, to get a feel for the system. It s important to release the puck BEFORE it has left the ramp, to ensure that you re not interfering with it during its flight. Turn on the spark timer and select a sparking interval (dt) of 50 milliseconds, and make a note of it in your log. The timing of the sparks is accurate to ±1%, which corresponds to ±0.5 ms on the 50 ms setting. MAKE SURE THAT YOU ARE HOLDING THE PUCK BY ITS VINYL TUBE and you are NOT TOUCHING THE TABLE OR THE OTHER PUCK OR YOU WILL GET A SHOCK! Hold down the two buttons that start the spark and launch the puck as you have done in the trials. NOTE: Each student should make her/his own set of tracings and calculations. Otherwise the workload ends up being uneven, and you won t learn as much. Turn the sheet of paper over and transfer the line CD by tracing through the paper. Draw a line parallel to CD along the line of spark points which were recorded while the puck was still in contact with the ramp and mark the position of the centre of the puck just as the puck left the ramp. This is the point (0,0). Carefully draw through (0,0) parallel to DE.

5 5 Now you can make the measurements that are requested in the worksheet (Tables 1-3) and use the recorded values to calculate the requested quantities (Table 4).

6 6 Name: Student ID Partners Name Date WORKSHEET: To be handed in: 1. Constants: VALUE UNCERTAINTY Radius of puck (This will effect your choice of (0,0)) Time between sparks Launch angle, α Length of air table between feet (hypotenuse) Height of block used to raise table (opposite) 2. The number of spark timer points until the puck: NUMBER OF SPARKS FROM ORIGIN UNCERTAINTY Is back to y = 0 (N R ) Reaches y max (N max ) Reaches the final point (N final ) NOTE: these numbers will not, in general, be whole numbers. You will have to estimate fractions of intervals, with appropriate uncertainties.

7 7 3. Displacement measurements: DISPLACEMENT UNCERTAINTY Average s in one interval at the instant the puck leaves the ramp The horizontal range, R y max The (x, y) co-ordinates selected final point NOTE: s is the distance travelled by the puck in that time interval. 4. Analysis Table: VALUE UNCERTAINTY v = s/dt v = R/(cosα t R ) a (using R and equation 6) a (using y max and equation 7) x f (using equation 4, and average a) y f (using equation 5, and average a) sin θ, inclination of air table = opposite/hypotenuse Be sure to record the systematic error for every measurement you take! For this experiment it is the systematic errors which determine the final uncertainty in your results. Consult the notes on error analysis if you re confused as to how to propagate your uncertainties. 5. Do the two values for v agree within experimental error? Which is more precise? 6. Compare x f, y f to their measured values. 7. Determine the theoritical value of the acceleration, a and its uncertainty, using a = g sin θ, where θ is the inclination of the air table. 8. Discuss the agreement of the different determinations of v, of a and of x f and y f, taking the experimental errors into account. Can you find any evidence of friction in your results?