Determining the Relationship Between the Range and Initial Velocity of an Object Moving in Projectile Motion

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1 Determining the Relationship Between the Range and Initial Velocity of an Object Moving in Projectile Motion Sadaf Fatima, Wendy Mixaynath October 07, 2011 ABSTRACT A small, spherical object (bearing ball) is released from various heights on an inclined ramp and permitted to roll down and then fall into projectile motion. The range of the object after it experienced projectile motion was recorded. A log-log graph of the height and the range was plotted to show the relationship between the initial velocity of the object and its range. 1 INTRODUCTION Objects moving in projectile motion experience a displacement that can be described in two components: the vertical displacement, and the horizontal displacement. The horizontal displacement can also be referred to as the range. The range of an object moving in projectile motion depends on the following factors: Angle of projection (with respect to the horizontal) Initial velocity of object Maximum height reached during projection (vertical displacement) This lab is being conducted to determine the relationship between the initial velocity and the range of an object experiencing projectile motion. Changing the initial velocity of the bearing ball at the moment it is released is not possible while working within the limitations on the apparatus. Due to this condition, it is much simpler to vary the height at the point from where the ball is released on the ramp. The velocity can be calculated later using equation (5). The angle of projection and the vertical displacement is kept constant. The range is the dependent variable, and the height is the independent variable. The mass of the bearing ball is negligible. Using equation (1) and (6), the hypothesis was determined to be that if the velocity is directly proportional to the range, then height of the ramp at the point the ball was released has a square root relationship with the range.

2 Vertical Displacement (Δy) PAWS THEORY θ = 0 Height (h) Horizontal Displacement/Range (Δx) Figure 1: Bearing ball undergoing projectile motion after rolling down from inclined ramp at an angle of 0 Figure 1 shows an object moving in projectile motion. The range can be determined using: and 2) where is the range; is the horizontal component of the object s initial velocity ( ; is the time interval for the object while it is in projectile motion; and is the angle of projection (with respect to the horizontal). In this lab, the angle was kept constant at θ= 0. When (1) and (2) are combined with the above condition: The time interval (t) in (3) can be eliminated by using the following equation: (4) where is the vertical displacement. The initial velocity in (3) can be eliminated by using the following equation: (5) 2 P a g e

3 where (g) is acceleration of gravity; and (h) is the height of the ramp at the point from which the ball is released Substituting (4) and (5) in (3), yields the equation: In order to linearize the data, the log is taken: (7) Equation (7) shows that the expected slope of the data when graphed will be 0.5, and the y- intercept will be. 3 Experiment Apparatus: 1. Hot Wheels Ramps 2. Retort Stand 3. Clamps 4. Bearing ball 5. Meter stick 6. Carbon paper 7. Clamps 8. White Paper 9. Duct Tape Procedure: (See Figure 2 for a visual representation of the procedure) 1. Set up the retort stand on a table and attach the Hot Wheels ramps to the stand, making sure to secure them in place by the clamps. 2. Tape down at least 3 cm of the other end of the ramp down to the table so that it makes an angle of 0 with the horizontal. 3. Using the meter stick, measure and mark 5 points on the ramp. These will be the points where you release the ball (Δh). 4. Using the metre stick, measure and record the distance from the end of the ramp (that is taped to the table) to the ground. This will be your vertical displacement. ( y) 5. Do a test run. Place the bearing ball on one of the points on the ramp that were marked in Step 3. 3 P a g e

4 Vertical Displacement (Δy=0.76m) PAWS Put a small ruler or something sturdy in front of the ball to keep it from rolling down. Lift up the ruler and to allow the ball to roll down the ramp. (This will ensure the bearing ball starts with an initial speed of zero) 7. Observe where the bearing ball lands on the ground. Place the carbon paper on the floor where the ball landed. Place a sheet of white paper under. Tape the carbon paper in place. 8. Now place the bearing ball back onto the same position on the ramp. Repeat step 6. Use a metre stick to measure and record the distance from the point on the ground perpendicular to the end of the ramp that the ball dropped from to the point or marking that the ball has made on the carbon paper. This is the horizontal displacement/range ( x). 9. Repeat for another 2 trials. Once done all 3 trials for a certain point on the ramp (Δh), calculate and record the average range of all 3 trials. 10. Repeat steps 5-9, for all the different points marked on the ramp. Figure 2: Diagram of Apparatus and Setup 4 Data 5 Conclusion Bearing Ball Ramp Height (Δh) Carbon Paper Horizontal Displacement/Range (Δx) 4 P a g e

5 Average position of ball on ramp when released( h) Initial velocity (m/s) Range ( x) Range Trial 1 Range Trial 2 Range Trial Vertical Displacement ( y) 0.76 Table 1: Experimental results for the range when bearing ball released from various heights on ramp. All units in metres unless otherwise stated. 4 Data The results show that as the ball was positioned higher up the ramp, the initial velocity increased. As the initial velocity increased, the range increased as well. Average x h (m) Log ( h) (m) Log( x) Table 2: Log Calculations for Height and Range The log of height and range summarized in Figure 2 were calculated in order to construct a loglog graph so that the relationship between height and range could be linearized. Figure 3 shows the square root relationship between the height of the ball when released on the ramp and the range of the ball after experiencing projectile motion. The linear regression yields an equation with a slope of and a y-intercept of P a g e

6 Log Base 10 of Range PAWS 2011 Linearized Relationship Between Range of Bearing Ball Experiencing Projectile Motion and Height of Ramp at Point When Ball Released Log Base 10 of Height of Ramp When Ball Released -0.4 Figure 3: Log-Log graph shows the relationship between the height of the ball when released on the ramp and the range the ball experienced. Equation of trendline: x = h Conclusions The experimental results summarized in Table 1 proved the hypothesis. The range of the ball increased as its initial velocity increased. This means that the range is indeed proportional to the initial velocity of an object. Figure 3 shows the square root relationship between the range and the height. The slope of the linear regression is 0.516, which is close (but not exact) to the theoretical result of 0.5. The y-intercept is The theoretical result was that the y-intercept would be is 3.2%. or, This means that errors must have occurred in the lab. The percent error A systematic error that could have occurred was failing to measure if the angle of the projection was actually 0.If the angle was not exactly zero, then it would change the horizontal component of the initial velocity, which in turn would change the results of the range measured. In future instances, the angle can be verified to be 0 by calculating the vertical distance from the top of the retort stand to the table, and the horizontal distance from the base of the retort stand to the end of the ramp where it is taped to the table. Then, trigonometry can be applied by finding the 6 P a g e

7 inverse tangent of the measured dimensions. Figure 4 is a visual representation of this. The percent error in this lab is low; if the angle was not exactly 0, it was close to it. Opposite Adjacent Figure 4: Verifying Figure 5: Positioning the second retort stand closer to the end of the table Another systematic error that may have played a part in reducing the accuracy of the results is the positioning of the retort stand when securing the ramp in place. Figure 2 shows that the retort stand was placed far away from the point where the ball dropped from. This caused the ramp to shake and wobble from the middle because there was no support there. In future instances, it would be best to position a second retort stand closer to the end of the table where the ball will drop from (as shown in Figure 5). This will provide support the ramp with support and reduce the shakiness of the ramp when the rolls down. 6 Acknowledgments Thank you to Phatdara Chhun for helping with the design and setup of the apparatus and for taking part during the experiment. Thank you to Mr. Murzaku for his notes and assistance throughout this lab. 7 P a g e

Projectile Motion. Equipment

Projectile Motion. Equipment rev 05/2018 Projectile Motion Equipment Qty Item Part Number 1 Mini Launcher ME-6800 1 Metal Sphere Projectile 1 and 2 Meter Sticks 1 Large Metal Rod ME-8741 1 Small Metal Rod ME-8736 1 Support Base ME-9355