3.1 Analysis in the absence of air resistanceIgnoring air resistance, Newton's equations of motion for an object under the influence of gravityalone are quite straightforward. We're also ignoring the fact that the gravitational pull on anobject above the earth depends upon its height above the earth. We take g = 9.81 m/s².y(1) = -g, *() = 0.(1)1. Solve these two second order differential equations (1) for x(t) and y(t) and useMATLAB or Excel to plot the (x,y) trajectory obtained for angle of inclination 8 = 35°.Observe that x(0) and y(0) were determined by the set-up of my measurements and areknown to be 0 and .18 m respectively. The remaining two initial conditions x'(0) and y'(0)are given byx'(0) = v cos 0,y'(8) = v sin 0,Where v is the initial velocity with which the gun fires its darts.2. Use the experimental data from Section 2 to determine the initial velocity with which thedarts are fired.3. Determine the distance, d(8), your object travels as a function of 0, and compute theangle that maximizes its distance. Your answer should be in degrees and accurate totwo decimal places. Plot this function in MATLAB or Excel and compare it with theexperimental values. Discuss the discrepancies.4. Compute an error, E, for your model based on the sum of squared errors,17E -(dexperimentat – dmodet)² 1 OverviewBallistics is the science of projectile motion and impact, phenomena well described byNewtonian mechanics. The number of applications of this type of analysis is staggering, rangingfrom such mundane issues as automobile accident simulations and optimal golfing to the criticalstudies of missile defense and space exploration. Somewhat less dramatically, in this project wewill use Newtonian mechanics to describe the flight of a sponge dart, light enough so that airresistance will play a critical role.2 Experimental DataThe data for this project was collected by firing sponge darts from a toy gun ($3.99, WalMart).The table below shows a set of measurements for distance traveled (by the projectile) versusangle of inclination of the gun, taking angles of inclination 5, 10, 15, . 85 degrees. The dartswere fired from a height of . 18 meters.Angle of 5InclinationDistance10152025303540454.375.236.957.848.178.698.818.998.95TraveledAngle ofInclinationDistance 8.83Traveled50556065707580858.197.847,126.385.083.342.13The time it took for a dart fired straight up from a height of .39 meters to hit the ground wasfound to be: 2.13 seconds. On the other hand, the time it took one of the darts to fall 4.06meters was found to be: .95 seconds. These fairly simple measurements will suffice for theassignments in this project. For the assignments in Sections 3.1 and 3.2, models will be basedentirely on these final two experiments, and the table of angles of inclination and distancestraveled will only be used for evaluation of the model.

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Ignoring air resistance, Newton's equations of motion for an object under the influence of gravity alone are quite straightforward. We're also ignoring the fact that the gravitational pull on an object above the earth depends upon its height above the earth. We take g = 9.81 m/s². y(t) = -9. x(1) = 0. (1) 1. Solve these two second order differential equations (1) for x(t) and y(t) and use MATLAB or Excel to plot the (x,y) trajectory obtained for angle of inclination 0 = 35°. Observe that x(0) and y(0) were determined by the set-up of my measurements and are known to be 0 and .18 m respectively. The remaining two initial conditions x'(0) and y'(0) are given by x'(0) = v cos 0, y'(8) = v sin 8, Where v is the initial velocity with which the gun fires its darts.

Ignoring air resistance, Newton's equations of motion for an object under the influence of gravity alone are quite straightforward. We're also ignoring the fact that the gravitational pull on an object above the earth depends upon its height above the earth. We take g = 9.81 m/s². y(t) = -9. x(1) = 0. (1) 1. Solve these two second order differential equations (1) for x(t) and y(t) and use MATLAB or Excel to plot the (x,y) trajectory obtained for angle of inclination 0 = 35°. Observe that x(0) and y(0) were determined by the set-up of my measurements and are known to be 0 and .18 m respectively. The remaining two initial conditions x'(0) and y'(0) are given by x'(0) = v cos 0, y'(8) = v sin 8, Where v is the initial velocity with which the gun fires its darts.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Gravitational force

In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.

Acceleration Due to Gravity

In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.

Question

3.1 Analysis in the absence of air resistance
Ignoring air resistance, Newton's equations of motion for an object under the influence of gravity
alone are quite straightforward. We're also ignoring the fact that the gravitational pull on an
object above the earth depends upon its height above the earth. We take g = 9.81 m/s².
y(1) = -g, *() = 0.
(1)
1. Solve these two second order differential equations (1) for x(t) and y(t) and use
MATLAB or Excel to plot the (x,y) trajectory obtained for angle of inclination 8 = 35°.
Observe that x(0) and y(0) were determined by the set-up of my measurements and are
known to be 0 and .18 m respectively. The remaining two initial conditions x'(0) and y'(0)
are given by
x'(0) = v cos 0,
y'(8) = v sin 0,
Where v is the initial velocity with which the gun fires its darts.
2. Use the experimental data from Section 2 to determine the initial velocity with which the
darts are fired.
3. Determine the distance, d(8), your object travels as a function of 0, and compute the
angle that maximizes its distance. Your answer should be in degrees and accurate to
two decimal places. Plot this function in MATLAB or Excel and compare it with the
experimental values. Discuss the discrepancies.
4. Compute an error, E, for your model based on the sum of squared errors,
17
E -
(dexperimentat – dmodet)²

1 Overview
Ballistics is the science of projectile motion and impact, phenomena well described by
Newtonian mechanics. The number of applications of this type of analysis is staggering, ranging
from such mundane issues as automobile accident simulations and optimal golfing to the critical
studies of missile defense and space exploration. Somewhat less dramatically, in this project we
will use Newtonian mechanics to describe the flight of a sponge dart, light enough so that air
resistance will play a critical role.
2 Experimental Data
The data for this project was collected by firing sponge darts from a toy gun ($3.99, WalMart).
The table below shows a set of measurements for distance traveled (by the projectile) versus
angle of inclination of the gun, taking angles of inclination 5, 10, 15, . 85 degrees. The darts
were fired from a height of . 18 meters.
Angle of 5
Inclination
Distance
10
15
20
25
30
35
40
45
4.37
5.23
6.95
7.84
8.17
8.69
8.81
8.99
8.95
Traveled
Angle of
Inclination
Distance 8.83
Traveled
50
55
60
65
70
75
80
85
8.19
7.84
7,12
6.38
5.08
3.34
2.13
The time it took for a dart fired straight up from a height of .39 meters to hit the ground was
found to be: 2.13 seconds. On the other hand, the time it took one of the darts to fall 4.06
meters was found to be: .95 seconds. These fairly simple measurements will suffice for the
assignments in this project. For the assignments in Sections 3.1 and 3.2, models will be based
entirely on these final two experiments, and the table of angles of inclination and distances
traveled will only be used for evaluation of the model.

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