Projectile Motion. Figure 1. The system of coordinates for the projectile motion.

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1 Projectile Motion (1) Introduction and Theory: Consider a projectile motion of a ball as shown in Fig. 1. At t = 0 the ball is released at the position (0, y0) with horizontal velocity vx. Figure 1. The system of coordinates for the projectile motion. Because of the influence of a constant gravitational acceleration in the Y direction, the x and y components of motion, according to Newton s laws of motion, are governed by the following equations: x = vx * t / 1 / y = y0 ½ * g * t 2 / 2 / where g = 9.81 m/sec 2 is the gravitational acceleration.

2 This set of equations describes a kinematics model for projectile motion based on Newton s first and second laws. There are two important facts to be noticed: (a) the distance the ball travels in the X direction is directly proportional to the time of flight; which is due to the fact that there is no force acting upon the ball in that direction, and (b) the distance the ball travels in the Y direction is proportional to the square of the time of flight, with the proportionality constant equal to one half of the gravitational acceleration. Text Reference: Wolfson 2.3. (2) Objectives: To test the validity of the kinematics model based on Newton s laws for simple projectile motion. (3) Equipment: Inclined plane; guided track; Science Workshop interface; photogates; time of flight accessory; ball; meter stick. accessory Figure 2. The experimental setup for Projectile Motion lab. (4) Procedure:

3 The experimental setup is depicted in Fig. 2. The ball, which is initially at rest at some position on the inclined V-shaped track, is released. It travels a certain distance along the horizontal U-shaped track for an amount of time t 12 (t 2 -t 1 ). The velocity is measured as it passes through the gate at position 2. This could be compared with a velocity derived from the time, t12, to travel the distance between points 1 and 2. The ball then falls as a projectile on the time of flight accessory after time t 23 (t 3 t 2 ). The point of impact is recorded by placing a piece of carbon paper on top of a regular paper taped carefully to the time of flight accessory. Open the pre-set experiment file PHY113Projectile to record time intervals t 12 and t 23 as well as the initial horizontal velocity of the projectile. Figure 3. A sample data table for projectile motion in Data Studio.

4 Press Start and let the ball roll and land on the Time-of-flight Accessory. Do not stop the recording. Bring the ball back to the same release point and let it go again. Each time the ball passes through the photogates and hits the time-of-flight plate a new set of data is added to the table. To finish one data run press Stop. Choose three start positions for the ball on the elevated part of the track and for each of them repeat the measurements of projectile motion 5 times. For each run compare the measured time of flight, t 23, (mean value +/- standard deviation of the mean) with the calculated one determined from eqn.2 and the known height of fall. Using the measured time of flight, t 23, and the measured initial velocity, v, compute for each run the horizontal range of projectile and compare it with the experimental one deduced from the cluster of positions recorded on the paper placed on the Time-of-flight Accessory. Any meaningful comparison should include the uncertainties. Final conclusion: Do your experimental results confirm the validity of eqn. 1 and eqn. 2? Discuss in your lab report the probable sources of error in your measurements.

5 PRE-LAB QUIZ PROJECTILE MOTION Name: Section: Consider a projectile launched horizontally with speed v from a height h The acceleration due to gravity is g.. 1. Assuming the projectile is launched over level ground and ignoring air resistance, what will be time-of-flight of the projectile? 2. What will be the range of the projectile? 3. If the error in v is σ v and the error in h is σ h, then what is the resulting error in the time-of-flight? Ignore any error in g. In your answer, you may use t as a symbol for the time-of-flight. 4. If the error in v is σ v and the error in h is σ h, then what is the resulting error in the range? Ignore any error in g. In your answer, you may use t as a symbol for the time-of-flight.