To quantify the speed of ball after it was fired we had measure different variables and apply physics with the help of mathematics. Just as important, we had to use a spring-loaded gun that had a block tied to four string connected to a pendulum as our experimental apparatus. As for measurements, we had to measure the height that the ball would take off from, the distance the ball would land after we shoot the ball out of the gun onto the floor, and the length of the strings; the mass was measured with a digital balance. The physics that helped us accomplish this experiment began when we shot the ball into the block until the end of the experiment when we shot the ball onto the floor because we used conservation of momentum theorem and the conservation of energy theorem to calculate the initial velocity of the ball and the time it takes the ball to hit the ground. For example, the initial velocity was calculated with the delta momentum technique and the conservation of energy technique and the time it took the ball to hit the ground was calculated with kinematics. Importantly, mathematics was needed for all of these calculations. Lastly, physical measurements for the distance that the ball traveled before it initially landed were done by using a carbon paper on top of a white piece of paper so it marked the impact site and a two meter ruler.
To begin with, we had to level the apparatus. This was done with the use of an iphone and an application named bubble level. After that, we weighed the block and the ball before we shot the ball into the block five times to calculate the final angle that the pendulum reached before we took an average of those values for our experimental angle value. Then, we derived a formula for the height the block would reach so we could use that value in our conservation of energy calculations. That value for the height was used to help find the final velocity of the ball and the block, so we could find the value for the initial velocity of the ball. Once we had the value for the initial velocity of the ball, we used kinematics to calculate where the ball would land after we shot it towards the floor. But, to verify that our distance value was reasonable we physically measured where the ball landed with a ruler after we marked the impact with the help of a carbon paper. Lastly, the physical measurement of the landing distance of the ball and the time calculation was used to calculate the initial velocity that corresponds to those values to compare with the mathematical calculations and physics.
This is an image of the spring loaded firing gun. Also, notice that I am measuring the height of the initial height of the ball with a two meter stick for further calculations.
This is an image of the physical measurement for the distance of initial impact of the ball when it hit the ground. Notice the carbon paper that was used to mark the impact site.
This is an image that illustrates the bob technique that we used to mark the starting position.
This image illustrates the given measured variables, the angle trial values, and the derived formula used to calculate the height of the block once it stopped on its rise.
This image illustrates the calculations for the conservation of momentum and energy values that were needed to calculate a theoretical initial and final speed of the ball and the block.
This image illustrates the calculations for the initial speed of the ball, the time it would take the ball hit the ground, and the distance the ball would travel in the air.
This image illustrates the uncertainty in mass and the angle, as well as the total mass and average measured angle.
In conclusion, we calculated theoretical values for the initial speed and distance traveled of the ball in first by taking some measurements, applying some physics, and some mathematics that resulted in the initial speed of 5.5 m/s and distance traveled of 2.47 m respectively. Then, we calculated experimental values for these variables with the use of physics and mathematics that resulted in an initial speed of 4.76 m/s and a total traveled distance of 2.14 m. Importantly, the absolute difference for the initial speed and the distanced traveled was 0.74 m/s and 0.33 m respectively. Hence, the values for the respective difference were fairly high given that they were 15% off the experimental values for initial speed and distance traveled in the air of the ball. As for causes of errors, there were probably more than one. For instance, there was probably some energy lost to friction when the ball went into the block because it seemed like the ball didn't go into the block without friction. So that could definitely cause an error in the final velocity of the block and the ball as well as the final height of the pendulum swing. Really friction would affect the whole theoretical and experimental calculations. Also, another reason for error was that we measured the height and distance traveled by the ball with an old simple ruler that could of easily been misread that would also throw calculations off. lastly, the balance that we used may of been off a little giving us a mass for the ball that was not correct that would also ruin further needed calculations. Although, the two values for theoretical and experimental initial velocity and distance traveled were in the same ball park so I think that shows that the physics and mathematics works, but the experiment was not controlled enough to show more accurate and precise values of interest.
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