Lab 5 (6th): Trajectories

Title:
Trajectories

Purpose:
This lab aims to produce a model to predict the trajectory of an object based on measurements of vertical and horizontal distance traveled with known launch angle and velocity.

Theory:
Ideally, an object experiencing projectile motion will have constant velocity in the horizontal direction while constantly accelerating downwards in the vertical direction. This means that its position in a 2-D plane can be represented as a function of its x position with its y position calculated through the kinematics equations. It is then possible to produce an approximate model describing the trajectory of an object in projectile motion if the initial x and y velocities are known and acceleration due to gravity is constant. This model would then be capable of predicting the position of an object at any point during its flight.

Apparatus:

• Aluminum "V" channel (2)
• Steel ball (1)
• Ring stand (1)
• Clamp (1)
• String with paperclip (1)
• Meter stick (1)
• Wooden plank
• Carbon paper
• Standard paper

Procedure:

Set up one aluminum V channel on a flat surface leading to a vertical drop. Position the other V channel at an angle so that the steel ball will accelerate down its length and continue along the other horizontal channel. Release the steel ball from the top of the V channel and note where it lands. Place a sheet of normal paper under a sheet of carbon paper at that location. Release the steel ball 5 more times. Using the paperclip, find the location directly under where the ball leaves the V channels. Measure the height from the V channel to the ground and measure the horizontal distance from the V channel to the dots created by the carbon paper. Place the wooden plank so that one end lies adjacent to the horizontal V channel and the other end rests on the ground. Measure the angle of the plank. Release the steel ball and note where it lands on the plank. Place a sheet of normal paper under a sheet of carbon paper at that location. Release the steel ball 5 more times and measure the distance of the dots from the upper end of the plank. Using the data available, derive an equation to predict where a steel ball would land on the plank given initial velocity and the angle of the plank.

Data and Graphs:

Photo of experimental setup

Table of distances traveled

Trial	Horizontal Δx (cm±0.01)	Δx on board (cm±0.01)
1	66,4	82,8
2	66,2	83,0
3	66,3	83,1
4	66,9	83,3
5	67,1	83,5
	Height (cm±0.01)	94,8
	Plank angle (±1°)	49

Calculations for velocity, predicted distance, and actual distance

Analysis:

From the data collected, an average value for horizontal distance was calculated. Using kinematic equations, a value of initial horizontal velocity and its uncertainty were calculated. An equation solving for distance down the board d was derived and a predicted value for d was calculated along with its uncertainty.

Conclusion:

Our predicted value of 0.820m with an uncertainty of 0.313m was near the actual value of 0.8314m found through measurement, leading to the conclusion that the model is adequate. The large uncertainty in predicted d value is mostly due to uncertainty in angle as the device used was not capable of producing highly accurate measurements. The experiment procedure itself is considered reliable as the idealized assumptions made would not have heavily affected the motion of the steel ball and measurement techniques remained consistent. Much of the error in our value is likely due to inaccuracies in measurement, particularly since the ruler had to be placed parallel to the V channels and the plank was difficult to keep steady at times.