Projectile Motion (Collisions in 2 Dimensions)

In this lab students will investigate the flight of a projectile when fired off a table horizontally. The horizontal distance which it flies before it hits the floor will be measured and compared to the theoretical distance it should fly. A comparison of velocities as the ball leaves the ramp will also be done. Then the ball will contact a target ball sitting motionless at the end of the ramp (not head on but at an angle). The horizontal and vertical components of the two balls will be resolved.

Procedure 1: (make all measurements in meters)

Attach the ramp to a table with clamps. Make sure the ball can roll down the center of the ramp and not hit anything. Make sure the ramp is horizontal at the table edge.
Place a ball at the end of the ramp. With a meter stick, measure the distance from the bottom of the ball to the floor. Record this height as h_t.
Place the ball at the highest point on the ramp where it roll down when released. Measure and record the change in height of the ball from this position and the position at the end of the ramp. Record this height as h_r.
Release the ball and note approximately where it lands. Repeat several times.
Tape a piece of white paper to the floor with its center approximately where the ball landed. Then place a piece of carbon paper over the white paper (do not tape carbon paper down).
Roll the ball down the ramp from the top of the ramp. When it hits the floor, it should hit the carbon paper and leave a dot on your white paper. Roll the ball at least 5 times down the ramp.
Tie a washer to a string and hang it underneath the end of the ramp to identify a position on the floor directly beneath the ramp (plumb bob). From this point on the floor, measure the distance to each of the black marks made by the ball on your white paper. Record these as d_t.
Do not remove the sheet of white paper taped down. It will be used in Procedure 2.

Analysis:

Find the average distance that each black mark was from under the ramp (average d_t).
Using the height of the ramp above the floor, find the time for the ball to fall from the ramp to the floor. Assume the vertical velocity is initially zero. t = Ö (2 x fall distance / g)
The ball moved horizontally the distance you found in #1 in the time you found in #2. Use these values to calculate the ball's velocity when it left the ramp. v = d/t
The potential energy which the ball has is changed into linear kinetic energy plus rotational kinetic energy. It can be shown that 2/7 of the energy is in the form of rotational kinetic energy and 5/7 is in the form of linear kinetic energy (kinetic energy associated with a mass moving in a straight line. Therefore: (5/7)mgh = (1/2)mv² Use this equation to find the velocity which the ball was going when it left the ramp. (hint: solve for v)
Calculate percent difference between the two answers for the velocity of the ball at the end of the ramp. Then calculate the potential energy of the ball at the top of the ramp. Calculate the percent difference between the potential energy there and 5/7 of the kinetic energy at the end of the ramp. Can differences be explained by measurement errors in the lab? What other sources of error are apparent? What effect on the lab would there be if the ramp were not horizontal at the lower end? How would you have to change your calculations?

Procedure 2: (again, make all measurements in meters)

Move the small ball holder (at end of ramp) so that it will hold a target ball at exactly the same height above the floor as the rolling ball at the end of the ramp. The rolling ball should not impact the target ball directly head on, but with a glancing blow. Place the target ball on the holder and roll the other ball. Locate on the floor where the two marbles hit. Do this several times and then tape down 2 separate sheets of target paper. Place sheets of carbon paper on these sheets (as in procedure 1). Locate the plumb bob position again and tape a piece of paper down so that the plumb bob points to a mark on one edge of the paper (see drawing below). Roll the ball several more times, this time making marks on the target paper. Do this 4 to 5 times.
Remove the carbon paper. On the sheet of paper taped down directly beneath the end of the ramp, draw a straight line out from the x (plumb bob mark) to the center of the set of dots (on each of the three sheets of paper). Measure the average distance out to the dots on the side pieces of paper.

Analysis:

We know the maximum velocity the single marble had at the end of the ramp (see calculation #3 in procedure 1). We know that the horizontal components of the two balls should cancel each other out and the vertical components of the two balls should add up to the maximum velocity. To see if this works:
Use the single sheet of paper taped down directly underneath the end of the ramp. Make certain you have drawn a line out the 5 dots for the single ball and also have drawn lines out to the impact zones of the two balls. We also know the average lengths of these lines. Using the fall time of the ball (from #2 in procedure 1) and these distances calculate the velocity of the balls. There will be three (one the same from procedure 1 and the two side velocities).
Measure the angles formed with the two side balls using your protractor.
Now we will calculate the horizontal components by resolving the x components of each triangle. Use the following formulas: first the right side triangle,

sin (angle q_right) = x / velocity of that ball solve for x

now for the left triangle,

sin (angle q_left) = x / velocity of that ball solve for x

are the two equal? close…

Now we will find the vertical components: first the right side triangle,

cos (angle q_right) = y / velocity of that ball solve for y

now for the left triangle,

cos (angle q_left) = y / velocity of that ball solve for y

now add those two y components… are they equal to the velocity you found in #3 in procedure 1?

Make a drawing showing the vectors involved. Include percent differences between two values for the initial velocity of the ball at the end of the ramp, the two X values, and the sum of the two Y values versus the initial velocity found by measuring V = d/t from procedure one.