I need to find the calculated initial speed and the angle for maximum range unde

Question

I need to find the calculated initial speed and the angle for maximum range under table 2. Also need to answer those 6 questions.

Introduction: In this lab you will investigate the motion of an object launched at various angles relative to the horizontal. Ignoring air resistance, we would expect that the vertical motion would be governed by a downward acceleration due to the force of gravity, while the horizontal motion would experience no acceleration and thus the horizontal component of velocity would remain constant. Using these ideas, and what you have already learned about constant acceleration motion in one dimension, we can derive some equations that allow us to calculate quantities such as the object's horizontal distance traveled, also called the range, the maximum height attained, and the launching angle that results in a maximum range for a given initial speed You will be using a projectile launcher that can be set to launch a projectile at various angles and a couple different speeds. You will first need to figure out the launch speed of your particular launcher, then, knowing the launch speed, you will fire the projectile at various angles (doing several trials at each angle) to determine the dependence of the range on the angle Warning!!! Projectiles can hurt! Be very careful while launching your projectile so you do not hit anyone, and be very careful while walking around so you do not get hit. To protect your eyes it is a good idea to wear safety goggles Part I: The Launch Speed: You will notice that the projectile launcher consists of a spring that can be compressed and locked in place. There are three different compression distances resulting in three different initial speeds as the projectile leaves the launcher. Use the middle compression setting unless you are too close to a wall, then use the first compression setting. Also, you might use this launcher in a later lab so record its identification number in the Data section. Set the launcher so that it launches your projectile horizontally off the edge of a table and onto the floor. Do a trial run to determine where the projectile will land. Be sure to protect any walls using one of the large boards provided by the instructor. Tape a piece of paper on the floor centered on the projectile's landing spot. Place (do not tape) carbon paper over this paper to mark subsequent landing spots. Now fire the projectile several times while recording the landing spot each time. After every firing be sure that the launcher hasn't moved. If it did move then you'll need to start over otherwise your distance measurements will be inconsistent. ldeally, all spots should be at the same location, but most likely they will be spread out. For each spot, measure the horizontal distance the projectile traveled. Calculate the average of these distances. Also, measure the vertical distance the projectile fell. Enter this data in Table 1 in the Data section How would you use this data to calculate the initial speed of the projectile? Hint: The initial velocity is purely horizontal and there is zero acceleration in the horizontal direction. A vertical velocity is gained by the acceleration due to gravity Part Il: Dependence of Range on Launch Angle: With the projectile launcher clamped to the table adjust the launch angle to 20°. Fire the projectile three or four times so that you can calculate an average horizontal distance traveled. Record this average distance in Table 2, in the Data section. You should also measure the projectile's initial height above the floor. Repeat the steps above for the remaining angles in Table 2

Explanation / Answer

1. from table 2

for launch angle theta, initial height h, average horizontal didstance is x

then

x = ucos(theta)*t ( where t is time of flight and u is initial launch speed)

also

0 = h + usin(theta)*t - 0.5gt^2

h = 0.5*g*x^2/u^2cos^2(theta) - usin(theta)*x/ucos(theta)

h = 4.905x^2/u^2cos^2(theta) - tan(theta)*x

hence

u^2cos^2(theta) = 4.905x^2/(1.14 + tan(theta)*x)

so from the data we can see

average initial launch speed, u = 3.736399089 m/s

from the graph we can see the angle of maximum range is aboutg 36 deg

1. theoretically

4.905x^2/u^2cos^2(theta) - xtan(theta) = h

solving for x

x = u^2*cos^2(theta)(tan(theta) - sqroot(tan^2(theta) + 19.62*h/u^2cos^2(theta)))/9.81

form the data of the question

x = 1.417cos^2(x)(tan(x)-sqrt(2.607tan^2(x)+ 1.607))

solving we get

angle of maximum range is about 50 deg which is differnet than the experimental value

2. the reason for difference in the expected and measurec value is because in theoretical calculation we neglect the effect of air resistance and hence we calculate for ideal case scenario andf not for practical case scenario

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