Projectile Motion - Revision history

Audreycho at 22:29, 21 April 2024

2024-04-21T22:29:37Z

← Older revision		Revision as of 18:29, 21 April 2024
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	Shot put is another such sport and a prime demonstration of fundamental projectile motion. In standard competitions, a shot put is 16lbs for men and 8.8lbs for women. Due to the large density of the ball and low velocity when thrown, air resistance minimally affects the trajectory of the path. The thrower must consider the height at launch, since the ball's landing height is lower, relatively. Thus, a 45 degree angle may not lead to an optimal throw. In practice, for most skilled shot putters, the optimal angle of release tends to be slightly below this. Since the starting height is above the landing height, throwers may put more forward velocity into their throws to achieve a greater horizontal distance!		Shot put is another such sport and a prime demonstration of fundamental projectile motion. In standard competitions, a shot put is 16lbs for men and 8.8lbs for women. Due to the large density of the ball and low velocity when thrown, air resistance minimally affects the trajectory of the path. The thrower must consider the height at launch, since the ball's landing height is lower, relatively. Thus, a 45 degree angle may not lead to an optimal throw. In practice, for most skilled shot putters, the optimal angle of release tends to be slightly below this. Since the starting height is above the landing height, throwers may put more forward velocity into their throws to achieve a greater horizontal distance!

			[[File:Shotput.png\|right]]

	===Application: cannonball firing===		===Application: cannonball firing===

Audreycho: /* Connectedness */

2024-04-21T22:16:25Z

Connectedness

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 ==Connectedness==
 Projectile motion has many applications in day-to-day life; every dropped, thrown, and launched object massive enough to disregard air friction is an example of projectile motion.
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 A bullet fired from a horizontal hunting rifle obeys the model of projectile motion, so by the time it reaches a given horizontal distance from the hunter, it will have fallen vertically by a predictable amount. The scope of a hunting rifle can be calibrated to specific distances. This angles the scope slightly below the horizontal to take into account the bullet drop; when the bullet reaches the horizontal distance of interest, it will have fallen exactly into the crosshair of the scope. The design and manufacture of the scopes and their calibration mechanisms require a quantitative understanding of projectile motion.
 ==History==

Audreycho at 21:34, 21 April 2024

2024-04-21T21:34:43Z

← Older revision		Revision as of 17:34, 21 April 2024
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	'''Edited by Sarah Kilgore November 2022'''		'''Edited by Sarah Kilgore November 2022'''

			'''Edited by Audrey Cho April 2024'''

	Motion in which an object is affected only by the constant force of gravity is referred to as projectile motion (and the object as a projectile). This scenario is necessarily a simplification, since one larger scales gravity is not constant as one moves away from the attracting object, and there is also usually air resistance acting upon the object. However, it is a useful approximation, and an instructive use of kinematics.		Motion in which an object is affected only by the constant force of gravity is referred to as projectile motion (and the object as a projectile). This scenario is necessarily a simplification, since one larger scales gravity is not constant as one moves away from the attracting object, and there is also usually air resistance acting upon the object. However, it is a useful approximation, and an instructive use of kinematics.

Skilgore7: I added the properties section as well as an additional diagram, and edited wording and specifics in other areas.

2022-11-29T21:06:59Z

I added the properties section as well as an additional diagram, and edited wording and specifics in other areas.

← Older revision		Revision as of 17:06, 29 November 2022
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	'''Edited by Richard Udall July 2019'''		'''Edited by Richard Udall July 2019'''

			'''Edited by Sarah Kilgore November 2022'''

	Motion in which an object is affected only by the constant force of gravity is referred to as projectile motion (and the object as a projectile). This scenario is necessarily a simplification, since one larger scales gravity is not constant as one moves away from the attracting object, and there is also usually air resistance acting upon the object. However, it is a useful approximation, and an instructive use of kinematics.		Motion in which an object is affected only by the constant force of gravity is referred to as projectile motion (and the object as a projectile). This scenario is necessarily a simplification, since one larger scales gravity is not constant as one moves away from the attracting object, and there is also usually air resistance acting upon the object. However, it is a useful approximation, and an instructive use of kinematics.
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	Projectile motion is a branch of classical mechanics which analyzes the motion of objects (projectiles) under the influence of the constant acceleration of gravity near the surface of the earth.<ref> https://www.physicsclassroom.com/class/vectors/Lesson-2/What-is-a-Projectile </ref> Often, the projectiles are propelled prior to being analyzed under the influence of gravity, giving them initial velocities that affect their trajectories. Two classic examples of projectile motion are the firing of a cannon and throwing a ball off a tower. An object is a projectile if it is subject only to the constant force of gravity. For real projectiles, other forces such as air resistance are usually present, but these make the mathematical model significantly more complicated, and fair estimations may be gotten when disregarding them for many cases (note, however, that the relative importance of these forces increases for a feather vs for a ball). Regardless of the direction of motion of the projectile, the free-body diagram of a projectile is always the same, and constant throughout its trajectory: a particle on which only gravity acts in a downward direction.		Projectile motion is a branch of classical mechanics which analyzes the motion of objects (projectiles) under the influence of the constant acceleration of gravity near the surface of the earth.<ref> https://www.physicsclassroom.com/class/vectors/Lesson-2/What-is-a-Projectile </ref> Often, the projectiles are propelled prior to being analyzed under the influence of gravity, giving them initial velocities that affect their trajectories. Two classic examples of projectile motion are the firing of a cannon and throwing a ball off a tower. An object is a projectile if it is subject only to the constant force of gravity. For real projectiles, other forces such as air resistance are usually present, but these make the mathematical model significantly more complicated, and fair estimations may be gotten when disregarding them for many cases (note, however, that the relative importance of these forces increases for a feather vs for a ball). Regardless of the direction of motion of the projectile, the free-body diagram of a projectile is always the same, and constant throughout its trajectory: a particle on which only gravity acts in a downward direction.

			===Properties===
			[[File:parabolic-motion.png\|right]]
			1. Motion is considered parabolic. <br />
			2. Horizontal component of acceleration is considered to be zero. <br />
			3. Vertical component of acceleration is constant because of gravity. <br />
			4. Horizontal and Vertical motion components are independent but share the same time. <br />


	===A Mathematical Model===		===A Mathematical Model===
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	===4. (Bonus)===		===4. (Bonus)===
	As a challenge, determine the angle at which the distance traveled by a projectile shot with a fixed velocity will be maximized, and prove that this is the maximum. Assume that the initial and final height will be the same. Hint: find the range of the projectile as a function of the launch angle, then use optimization (find the angle at which the derivative of this function with respect to angle is equal to zero).		As a challenge, determine the angle at which the distance traveled by a projectile shot with a fixed velocity will be maximized, and prove that this is the maximum. Assume that the initial and final height will be the same. Hint: find the range of the projectile as a function of the launch angle, then use optimization (find the angle at which the derivative of this function with respect to the angle is equal to zero).

	<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">		<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
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	<math> \vec{\mathbf{F}}_{normal} = (0,0,\mathbf{m}\|\mathbf{g}\|) = (0,0,9.8) N </math>		<math> \vec{\mathbf{F}}_{normal} = (0,0,\mathbf{m}\|\mathbf{g}\|) = (0,0,9.8) N </math>

	Now we compute friction. Friction will always be directed in the opposite direction of the velocity, so just as velocity has vector for <math> \vec{\mathbf{v}} = (3,4,0) \; m/s </math>, friction will have vector form <math> \vec{\mathbf{F}}_{frict} = \|\vec{\mathbf{F}}_{frict}\|(-3/5,-4/5,0) </math>, where <math> \|\vec{\mathbf{F}}_{frict}\| </math> is the magnitude of the frictional force, and the vector has been normalized, so that ~~the~~ its magnitude is equal to one (you should recognize 3,4, and 5 as a ~~pythagorean~~ triple). Therefore, we calculate		Now we compute friction. Friction will always be directed in the opposite direction of the velocity, so just as velocity has vector for <math> \vec{\mathbf{v}} = (3,4,0) \; m/s </math>, friction will have vector form <math> \vec{\mathbf{F}}_{frict} = \|\vec{\mathbf{F}}_{frict}\|(-3/5,-4/5,0) </math>, where <math> \|\vec{\mathbf{F}}_{frict}\| </math> is the magnitude of the frictional force, and the vector has been normalized, so that its magnitude is equal to one (you should recognize 3,4, and 5 as a Pythagorean triple). Therefore, we calculate

	<math> \|\vec{\mathbf{F}}_{frict}\| = \mu \|\vec{\mathbf{F}}_{normal}\| = 2.94 \;N </math>		<math> \|\vec{\mathbf{F}}_{frict}\| = \mu \|\vec{\mathbf{F}}_{normal}\| = 2.94 \;N </math>
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	'''Find Net Force'''		'''Find Net Force'''

	Because gravity is directed in the z direction, the force will have only a z component, which we may calculate:		Because gravity is directed in the z-direction, the force will have only a z-component, which we may calculate:

	<math> \vec{\mathbf{F}}_{net} = \vec{\mathbf{F}}_{g} = (0,\mathbf{m}\mathbf{g},0) = (0, (5\;kg)(9.8 \;\frac{m} {s^2}),0) = (0,49,0)\; N</math>		<math> \vec{\mathbf{F}}_{net} = \vec{\mathbf{F}}_{g} = (0,\mathbf{m}\mathbf{g},0) = (0, (5\;kg)(9.8 \;\frac{m} {s^2}),0) = (0,49,0)\; N</math>
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	<math> t = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{ -10 \; \frac{m}{s} \pm \sqrt{100 \; \frac{m^2}{s^2} + 1960 \frac{m^2}{s^2}}}{-9.8 \; \frac{m}{s^2}} </math>		<math> t = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{ -10 \; \frac{m}{s} \pm \sqrt{100 \; \frac{m^2}{s^2} + 1960 \frac{m^2}{s^2}}}{-9.8 \; \frac{m}{s^2}} </math>

	Notice the negative denominator, so it is the minus option ~~which~~ will be positive. Since negative times are unphysical, we discard the plus case and have that <math> t \approx 5.65 s </math>.		Notice the negative denominator, so it is the minus option that will be positive. Since negative times are unphysical, we discard the plus case and have that <math> t \approx 5.65 s </math>.

	'''Find Displacement in x'''		'''Find Displacement in x'''
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	c)What is the velocity of the ball as it crosses the posts?		c)What is the velocity of the ball as it crosses the posts?

	d)What was the ~~balls~~ maximum height?		d)What was the ball's maximum height?

	e)What was the force on the ball if the ball is .76 kg and the impact lasts for 34 ms?		e)What was the force on the ball if the ball is .76 kg and the impact lasts for 34 ms?
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	'''Find Max Time'''		'''Find Max Time'''

	The velocity at the max point is 0. Use equation from part A:		The velocity at the max point is 0. Use the equation from part A:

	<math> v_{max y} = v_{initial y} - g \Delta{t_{max}} = 0</math>		<math> v_{max y} = v_{initial y} - g \Delta{t_{max}} = 0</math>
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	==History==		==History==
	The history of projectile motion is closely entwined with the history of [[Kinematics]], and so in many ways the history of one is the history of the other. A few special notes may be made about projectiles though. Aristotle was one of the first to write extensively about the motion of objects, and made a number of claims. First, he argued that heavy objects (like stones) fall because they are made primarily of earth, which is attracted to other earth, whereas smoke rises because it is made of air, and is attracted to other air. <ref>http://www.pas.rochester.edu/~blackman/ast104/aristotle_dynamics13.html </ref> He also believed that objects move only due to propelling forces, leading him to conclude that the air itself pushes objects forward, through some form of vacuum interaction. <ref> http://galileo.rice.edu/lib/student_work/experiment95/paraintr.html </ref> <ref> http://www.pas.rochester.edu/~blackman/ast104/aristotle_dynamics13.html </ref>		The history of projectile motion is closely entwined with the history of [[Kinematics]], and so in many ways, the history of one is the history of the other. A few special notes may be made about projectiles though. Aristotle was one of the first to write extensively about the motion of objects and made a number of claims. First, he argued that heavy objects (like stones) fall because they are made primarily of earth, which is attracted to other earth, whereas smoke rises because it is made of air, and is attracted to other air. <ref>http://www.pas.rochester.edu/~blackman/ast104/aristotle_dynamics13.html </ref> He also believed that objects move only due to propelling forces, leading him to conclude that the air itself pushes objects forward, through some form of vacuum interaction. <ref> http://galileo.rice.edu/lib/student_work/experiment95/paraintr.html </ref> <ref> http://www.pas.rochester.edu/~blackman/ast104/aristotle_dynamics13.html </ref>

	It was Galileo who recognized the flaws in this argument, and who instead put forth the idea of inertia, that objects will ''stay'' in motion unless acted upon by outside forces. <ref> http://ffden-2.phys.uaf.edu/211.fall2000.web.projects/J.%20Gentry%20and%20D.%20Arnold/phys211.html </ref> Galileo also identified the parabolic arcs characteristic of projectile motion.<ref> http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion2.html </ref> He did this, and identified the separability of the horizontal and vertical motion, by rolling balls off of inclined planes and then over a ledge, and noting the resultant trajectories.		It was Galileo who recognized the flaws in this argument, and who instead put forth the idea of inertia, that objects will ''stay'' in motion unless acted upon by outside forces. <ref> http://ffden-2.phys.uaf.edu/211.fall2000.web.projects/J.%20Gentry%20and%20D.%20Arnold/phys211.html </ref> Galileo also identified the parabolic arcs characteristic of projectile motion.<ref> http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion2.html </ref> He did this, and identified the separability of the horizontal and vertical motion, by rolling balls off of inclined planes and then over a ledge, and noting the resultant trajectories.
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	===Aristotle===		===Aristotle===
	The first findings of Projectile Motion were contributed by Aristotle. He believed that a projectile modeled by a canon ball travels in a straight line until it lost its 'impetus'. When it lost its 'impetus' which ran out ~~after~~ time, due to a restive force, the projectile would immediately drop and fall to the ground. For objects projected over short distances, at first observation this can visually appear to be true.		The first findings of Projectile Motion were contributed by Aristotle. He believed that a projectile modeled by a canon ball travels in a straight line until it lost its 'impetus'. When it lost its 'impetus' which ran out over time, due to a restive force, the projectile would immediately drop and fall to the ground. For objects projected over short distances, at first observation this can visually appear to be true.

	===Galileo===		===Galileo===
	Galileo's findings were motivated by the Renaissance. Through as series of experiments, he discovered that projectile motion is a result of free fall motion along the y-axis and uniform (constant velocity) motion along x-axis. This model became well-known and led to new insights on projectile motion, such as its parabolic shape. It is still accepted today.		Galileo's findings were motivated by the Renaissance. Through a series of experiments, he discovered that projectile motion is a result of free fall motion along the y-axis and uniform (constant velocity) motion along the x-axis. This model became well-known and led to new insights into projectile motion, such as its parabolic shape. It is still accepted today.

	One experiment Galileo performed in developing this model consisted of dropping two objects from equal heights: one from rest, and one with an initial horizontal velocity. Both struck the ground at the same time because both had the same initial vertical velocity. The horizontal component of velocity does not affect its vertical motion because the components of motion are independent.		One experiment Galileo performed in developing this model consisted of dropping two objects from equal heights: one from rest, and one with an initial horizontal velocity. Both struck the ground at the same time because both had the same initial vertical velocity. The horizontal component of velocity does not affect its vertical motion because the components of motion are independent.
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	===Isaac Newton===		===Isaac Newton===

	Isaac Newton contributed to the idea of projectile motion with his idea of a cannon on a mountaintop, now known as Newton's cannonball. Newton hypothesized that a cannon on top of a high mountain could fire a ball with varying initial speeds for varying trajectories. A cannonball simply dropped off the mountain would merely return to earth, but if that cannonball were fired from a cannon then it would return to Earth in an arc, or parabolic shape. This would be considered projectile motion. Neither of these scenarios ~~were~~ new or particularly innovative, but Newton also imagined a third scenario in which the cannonball ~~were~~ launched at great speed. In this scenario, he imagined that the ball would travel so far before curving downward that the earth would curve underneath it due to its spherical shape, resulting in a circular orbit. In doing so, Isaac Newton became one of the first to imagine universal gravity and to come up with a cause for orbital motion. It is worth noting that Newton's contribution does not strictly fall under the category of projectile motion as it is defined on this page because in the case of a cannonball orbiting the earth, gravity is not constant and downward, but changes its direction depending on the cannonball's position to point towards the center of the earth.		Isaac Newton contributed to the idea of projectile motion with his idea of a cannon on a mountaintop, now known as Newton's cannonball. Newton hypothesized that a cannon on top of a high mountain could fire a ball with varying initial speeds for varying trajectories. A cannonball simply dropped off the mountain would merely return to earth, but if that cannonball were fired from a cannon then it would return to Earth in an arc or parabolic shape. This would be considered projectile motion. Neither of these scenarios was new or particularly innovative, but Newton also imagined a third scenario in which the cannonball was launched at great speed. In this scenario, he imagined that the ball would travel so far before curving downward that the earth would curve underneath it due to its spherical shape, resulting in a circular orbit. In doing so, Isaac Newton became one of the first to imagine universal gravity and to come up with a cause for orbital motion. It is worth noting that Newton's contribution does not strictly fall under the category of projectile motion as it is defined on this page because, in the case of a cannonball orbiting the earth, gravity is not constant and downward, but changes its direction depending on the cannonball's position to point towards the center of the earth.
	-->		-->

Richard.Udall: /* 4. (Bonus) */

2019-07-29T20:00:55Z

4. (Bonus)

← Older revision		Revision as of 16:00, 29 July 2019
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	<math> \frac{\text{d}^2x}{\text{d}\theta^2}\biggr{\|}_{\theta = \pi/4} = -\frac{4v_0^2}{g} </math>		<math> \frac{\text{d}^2x}{\text{d}\theta^2}\biggr{\|}_{\theta = \pi/4} = -\frac{4v_0^2}{g} </math>

			Since this is strictly negative, we know that this point is a maximum, which means that <math> \theta = \pi/4 = 45^{\circ} </math> is the angle at which the projectile will travel a maximum distance.

	</div></div>		</div></div>

Richard.Udall: /* 4. (Bonus) */

2019-07-29T19:58:33Z

4. (Bonus)

← Older revision		Revision as of 15:58, 29 July 2019
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	and substitute this in to <math> x_f </math> to arrive at		and substitute this in to <math> x_f </math> to arrive at

	<math> x_f = \frac{2v_0^2 \sin(\theta)\cos(\theta)}{g} = \frac{ v_0^2\sin(2\theta)}{g}</math>		<math> x_f = \frac{2v_0^2 \sin(\theta)\cos(\theta)}{g} = \frac{v_0^2\sin(2\theta)}{g} </math>

	Now we optimize the solution, finding that		Now we optimize the solution, finding that
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	<math> \frac{\text{d}^2x}{\text{d}\theta^2} = - \frac{4v_0^2\sin(2\theta)}{g} </math>		<math> \frac{\text{d}^2x}{\text{d}\theta^2} = - \frac{4v_0^2\sin(2\theta)}{g} </math>

			then plugging <math> \theta = \pi/4 </math> to arrive at

			<math> \frac{\text{d}^2x}{\text{d}\theta^2}\biggr{\|}_{\theta = \pi/4} = -\frac{4v_0^2}{g} </math>

	</div></div>		</div></div>

Richard.Udall: /* 4. (Bonus) */

2019-07-29T19:56:33Z

4. (Bonus)

← Older revision		Revision as of 15:56, 29 July 2019
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	<math> x_f = \frac{2v_0^2 \sin(\theta)\cos(\theta)}{g} = \frac{ v_0^2\sin(2\theta)}{g}</math>		<math> x_f = \frac{2v_0^2 \sin(\theta)\cos(\theta)}{g} = \frac{ v_0^2\sin(2\theta)}{g}</math>

			Now we optimize the solution, finding that

	~~</div></div>~~















			<math> \frac{\text{d}x}{\text{d}\theta} = 0 = \frac{2v_0^2\cos(2\theta)}{g} </math>

			This allows us to solve for <math> \theta </math> and conclude that

			<math> \theta = \frac{\pi}{4} </math>

			We may check that this is indeed maximization, calculating that

			<math> \frac{\text{d}^2x}{\text{d}\theta^2} = - \frac{4v_0^2\sin(2\theta)}{g} </math>

			</div></div>

	<b>Additional examples of projectile motion can be found on the [[Kinematics]] page.</b>		<b>Additional examples of projectile motion can be found on the [[Kinematics]] page.</b>

Richard.Udall: /* 4. (Bonus) */

2019-07-29T19:51:23Z

4. (Bonus)

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 ===4. (Bonus)===
-As a challenge, determine the angle at which the distance traveled will be maximized, and prove that this is the maximum. Hint: find the range of the projectile as a function of the launch angle, then use optimization (find the angle at which the derivative of this function with respect to angle is equal to zero).
+As a challenge, determine the angle at which the distance traveled by a projectile shot with a fixed velocity will be maximized, and prove that this is the maximum. Assume that the initial and final height will be the same. Hint: find the range of the projectile as a function of the launch angle, then use optimization (find the angle at which the derivative of this function with respect to angle is equal to zero).
 <b>Additional examples of projectile motion can be found on the [[Kinematics]] page.</b>

Richard.Udall: /* See also */

2019-07-29T19:13:19Z

Richard.Udall: /* See also */

2019-07-29T19:12:56Z

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	*[[Frame of Reference]]		*[[Frame of Reference]]
	*[[Iterative Prediction]]		*[[Iterative Prediction]]
			*[[Terminal Speed]]

	===External links===		===External links===

← Older revision		Revision as of 15:12, 29 July 2019
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	== See also ==		== See also ==
			*[[Kinematics]]
	[[Kinematics]]		*[[Velocity]]
			*[[Acceleration]]
	[[Velocity]]		*[[Frame of Reference]]
			*[[Iterative Prediction]]
	[[Acceleration]]

	[[Frame of Reference]]

	[[Iterative Prediction]]

	===External links===		===External links===

	[http://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles/Projectile-Simulator Interactive Projectile Simulator]		*[http://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles/Projectile-Simulator Interactive Projectile Simulator]

	===Further Reading===		===Further Reading===