PhysicsLAB: Target Lab

Purpose: To predict the landing point of a projectile after it has rolled down a ramp.

Equipment needed: Each group needs: ramp, clamp, steel ball, target paper, carbon paper, meter stick, plumb line

Resource Lessons: 2D projectiles (horizontal release), Conservation of Energy.

Background

Remember that when analyzing two-dimensional projectile motion, the horizontal and vertical motions are independent of each other. Horizontally, projectiles in freefall travel at a constant velocity; while vertically, they experience uniform acceleration resulting in a classic parabolic trajectory. Our secret to working projectile problems was to build an H | V chart in which we delineated the properties of each dimension.

Horizontally, the only equation available to us was R = v_Ht, where v_H represents the projectile's constant horizontal velocity. Vertically, in the above illustration, the projectile's initial velocity equaled zero, since it was launched straight forward. Usually, in this situation, we let v_o = 0, a = -9.8 m/sec², and s = -h and then used the kinematics equation s = v_ot + ½at² to solve for the time that the projectile spent in the air.

Your goal in this experiment is to predict where a steel ball will land on the floor after having rolled down an incline plane. The final test of your measurements and computations will be to position a bull's-eye on the floor so that the ball lands in its center circle on the first attempt. Make sure that ALL measurements and calculations are reported with three significant figures.

Phase I: Calibrating Your Ramp

Step 1: Assemble your ramp. Make it as sturdy as possible so the steel ball rolls smoothly and consistently. The ramp should not sway or bend. Since the ball must leave the table horizontally, make sure that the horizontal part of the ramp is level with the surface of the table. The vertical height, h, of the ramp should be no less than 7 cm.

Step 2: Calculate the ball's horizontal velocity at the base of the ramp using conservation of energy principles. At the top of the ramp, if the ball is released from rest, it will only have potential energy, PE, which equals the product of its mass (in kilograms) times the acceleration due to gravity (9.8 m/sec²) and its height (in meters) above an arbitrary reference line. At the base of the ramp, the ball has translational kinetic energy, KE, which equals half the product of its mass (in kilograms) times the square of its velocity (in m/sec).

PE_top = KE_base
mgh = ½mv²
2mgh = mv²
√2gh = v

This velocity at the base of the incline will remain the ball's horizontal velocity when it leaves the table. Remember that you will need to consistently release the ball from the same height on the ramp as well as not put any pressure against the ramp that might result in it "springing" forward when the ball is released.

How high (in cm) was the back of your ramp (ruler) above the top of the table?

Show your calculations for the ball's horizontal velocity in the space provided below on your answer sheet. What will be your ball's ideal horizontal velocity (in m/sec) at the base of its ramp (ruler)?

Why did you not need to measure the ball's mass for the previous calculation?

Step 3: Using a plumb line, string, and meter stick measure and record here the vertical height of the lab table above the floor.

Height of table (in cm) =

Step 4: Using the appropriate equation from the background information given above, calculate the time, t, that the ball will take to fall from the base of the ramp on the table's surface to the floor.

t (in sec) =

Step 5: The range is the horizontal distance a projectile once it is leaves the table until it strikes the floor. Calculate the range of the ball. Show your equation and any necessary calculations used in predicting the ball's range.

R (in m) =

Teacher certification that you have calculated your experimental range.

Step 6: Now tape the center of the bull's-eye on the floor where you predict the ball will strike. When you are ready to release your ball, call your instructor over to witness your trial. Remember to make sure that the ball is released from the top of the ramp. Leaving the target paper taped to the floor, measure how far the ball struck from the center of the bull's-eye.

End of Phase I: Our ball missed the center of the bullseye by ___ cm.

Phase II: Reaching the bullseye

Step 7: Leaving the target paper in it's original location, measure the ball's actual range.

actual range: R (in m) =

The projectile missed the target because not all of its gravitational potential energy at the top of the ramp was converted to kinetic energy at the base of the ramp.

Step 8: To calculate the ball's exact energy content, you need to know its mass. Record its mass (in kg) in the blank provided.

Step 9: What is the ball's potential energy at the top of the ramp (in joules)?

Step 10: What is the ball's kinetic energy at the base of the ramp (in joules)

Step 11: How much mechanical energy was lost by the ball as it rolled down the ramp (in joules)

Step 12: Account for the form(s) of this lost energy.

The question now becomes how can more energy be added to the system to compensate for the energy lost as the ball rolled down the incline. Compensating for this loss will allow you attain your goal of the ball striking the bullseye.

Since you are not allowed to change the shape or size of your ramp, your ball's actual vH cannot be changed. Hence, the lost energy must be added to the system in the form of additional gravitational potential energy.

Step 13: Use the amount of energy lost in Step 12 to determine the extra height needed. Express your answer in meters.

Teacher certification that you have calculated your new height.

Step 14. After making the adjustments outlined above, call your instructor over to witness a second release of your ball.

End of Phase II: Our ball came within ___ cm of hitting the center of the bullseye!

Phase III: Verifying Your Results

Step 15: Using your original experimental range and the actual v_H found in the previous question, calculate the time needed in the air for the projectile to reach the bullseye.

t (in sec) =

Step 16: How high was the base of the ramp placed above the floor in your second trial?

height (in m) =

Step 17: Use kinematics to calculate the time the ball actually spent in the air during your second trial.

t = ___ seconds

Step 18: What is the percent different between your two times?

Conclusions

Conclusion #1: Did your modifications work? If not, what further adjustments would you now consider making?

Conclusion #2: Using energy principles as your guide, does the steel ball possess another type of energy (either as it rolled down the incline or during freefall) that you did not consider in your calculations? Explain.

Conclusion #3: Based on the ball's actual horizontal velocity and the final height of your ramp above the floor, use energy-methods to calculate the resultant impact velocity of the ball as it struck the ground.

V_impact (in m/sec) =