Kinematics - Revision history

Kcross37: /* Visuals */

2025-04-23T03:50:35Z

Visuals

← Older revision		Revision as of 23:50, 22 April 2025
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	Motion Diagram: Free Fall Example		Motion Diagram: Free Fall Example

	(Arrows shrinking going up, growing going down.)		(Arrows shrinking going up, growing going down.)








	Position vs. Time Graph:		Position vs. Time Graph:



	A curve if accelerating		A curve if accelerating
	Straight line if moving at constant velocity		Straight line if moving at constant velocity











	===Sources===		===Sources===

Kcross37 at 03:43, 23 April 2025

2025-04-23T03:43:15Z

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			====Kyler Cross – Spring 2025====

			Kinematics in One Dimension

			===Introduction===
			Kinematics are what describe the motion of objects without having to consider the forces causing the motion. In one-dimensional kinematics, the motion happens along a single straight line straight up or down(either horizontal or vertical).

			Understanding 1D motion allows the formation of vital foundation for being able to analyze more complicated motion in two or three dimensions.

			In this section, we will explore displacement, velocity, acceleration, and time, while using the kinematic equations used to solve problems involving constant acceleration.


			===Key Terms===

			Displacement (Δx)
			Change in position
			Equation: Δx = x_final − x_initial

			Displacement is the straight-line distance from the starting point to the ending point, along with direction.
			Positive displacement: movement in the positive direction (rightward or upward).
			Negative displacement: movement in the negative direction (leftward or downward).


			Average Velocity (v̄)
			Total displacement divided by total time
			Equation: v̄ = Δx / Δt
			Instantaneous Velocity (v)
			Velocity at a particular instant

			Average velocity relates to the entire journey:
			v̄ = (x_final − x_initial) / (t_final − t_initial)
			Instantaneous velocity is the speed and direction of an object at a specific moment. For example, when your car’s speedometer reads 65 mph, that’s its instantaneous velocity (assuming it is only going in one dimension in motion).


			Acceleration (a)
			Rate of change of velocity
			Equation: a = Δv / Δt

			Acceleration is the change in velocity over time.
			An object accelerates when it is speeding up, slowing down, or changing direction.
			a = (v_final − v_initial) / (t_final − t_initial)



			===The Kinematic Equations with Constant Acceleration===

			When acceleration is constant (including free fall), the following four equations apply:

			v = v₀ + at
			Δx = v₀t + (1/2)at²
			v² = v₀² + 2aΔx
			Δx = (v + v₀)/2 × t

			Where:
			v₀ = initial velocity
			v = final velocity
			a = acceleration
			t = time
			Δx = displacement

			==Common Example Scenarios==

			1. Free Fall
			Objects falling near Earth’s surface (ignoring air resistance) experience constant acceleration downward: a = -9.8 m/s²

			Example:

			If you throw a ball straight up at 15 m/s, how long does the ball take until it reaches the highest point?

			v = v₀ + at

			0 = 15 + (-9.8)t

			t ≈ 1.53 seconds


			2. Car Braking Problem
			A car moving at 20 m/s slams on the brakes and stops over 50 meters. What is the car’s acceleration?

			Given:
			v = 0
			v₀ = 20 m/s
			Δx = 50 m

			Use the equation:

			v² = v₀² + 2aΔx
			0 = (20)² + 2a(50)
			400 + 100a = 0
			a = -4.0 m/s²

			The car’s acceleration is -4.0 m/s² (decelerating).

			===Visuals===

			Motion Diagram: Free Fall Example

			(Arrows shrinking going up, growing going down.)








			Position vs. Time Graph:



			A curve if accelerating
			Straight line if moving at constant velocity











			===Sources===


			OpenStax Physics (Chapter 2: Kinematics)

			Georgia Tech PHYS 2211 Course Materials

			GlowScript Trinket Physics Simulations





	'''Edited by YorickAndeweg June 2019'''		'''Edited by YorickAndeweg June 2019'''

YorickAndeweg: /* A Mathematical Model */

2019-08-08T21:38:33Z

A Mathematical Model

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	===A Mathematical Model===		===A Mathematical Model===

	These are the standard ~~kinematics~~ equations that are often used to model motion in physics<ref> http://www.physicsclassroom.com/class/1DKin/Lesson-6/Kinematic-Equations </ref>. That said, other equations do exist (for example, calculus-based versions of the above equations that do not assume constant acceleration, as well equations that refer to the higher time derivatives of position such as jerk).		====The Kinematic Equations====

			These are the standard kinematic equations that are often used to model motion in physics<ref> http://www.physicsclassroom.com/class/1DKin/Lesson-6/Kinematic-Equations </ref>. That said, other equations do exist (for example, calculus-based versions of the above equations that do not assume constant acceleration, as well equations that refer to the higher time derivatives of position such as jerk).

	<ol>		<ol>

YorickAndeweg: /* A Mathematical Model */

2019-08-08T21:27:53Z

A Mathematical Model

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	<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (Typically this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>		<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (Typically this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>
	<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (Assumes constant acceleration. A derivation for this formula is available [[Derivation of Average Velocity\|here]].)</li>		<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (Assumes constant acceleration. A derivation for this formula is available [[Derivation of Average Velocity\|here]].)</li>
	<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (Assumes constant acceleration)</li>		<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (Assumes constant acceleration.)</li>
	<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (Assumes constant acceleration)</li>		<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (Assumes constant acceleration.)</li>

	</ol>		</ol>

YorickAndeweg: /* A Mathematical Model */

2019-08-08T21:26:36Z

A Mathematical Model

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	<ol>		<ol>
	<li><math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \cdot t</math></li>		<li><math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \cdot t</math></li>
	<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (~~typically~~ this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>		<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (Typically this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>
	<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (~~assumes~~ constant acceleration. A derivation for this formula is available [[Derivation of Average Velocity\|here]].)</li>		<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (Assumes constant acceleration. A derivation for this formula is available [[Derivation of Average Velocity\|here]].)</li>
	<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (~~assumes~~ constant acceleration)</li>		<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (Assumes constant acceleration)</li>
	<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (~~assumes~~ constant acceleration)</li>		<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (Assumes constant acceleration)</li>

	</ol>		</ol>

YorickAndeweg: /* A Mathematical Model */

2019-08-08T21:25:41Z

A Mathematical Model

← Older revision		Revision as of 17:25, 8 August 2019
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	<li><math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \cdot t</math></li>		<li><math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \cdot t</math></li>
	<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (typically this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>		<li><math>\vec{v}_f = \vec{v}_i + \vec{a}_{avg} \cdot t</math> (typically this equation is used in situations of constant acceleration, where <math>\vec{a}_{avg}</math> is that constant acceleration.)</li>
	<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (assumes constant acceleration)</li>		<li><math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math> (assumes constant acceleration. A derivation for this formula is available [[Derivation of Average Velocity\|here]].)</li>
	<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (assumes constant acceleration)</li>		<li><math>\vec{r}_f = \frac{1}{2} \vec{a} \cdot t^2 + \vec{v}_i \cdot t + \vec{r}_i</math> (assumes constant acceleration)</li>
	<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (assumes constant acceleration)</li>		<li><math>v_f^2 = v_i^2 + 2 \vec{a}(\vec{r}_f - \vec{r}_i)</math> (assumes constant acceleration)</li>

YorickAndeweg: /* See also */

2019-08-08T21:22:04Z

YorickAndeweg: minor edit fixing a reference to one of the 5 kinematic equations

2019-08-01T00:25:56Z

minor edit fixing a reference to one of the 5 kinematic equations

← Older revision		Revision as of 20:25, 31 July 2019
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	Usually when kinematic equations are used to solve problems, one-dimensional versions are used so that vector addition doesn't need to be performed. For example, equation 3 can be decomposed into x and y components as follows:		Usually when kinematic equations are used to solve problems, one-dimensional versions are used so that vector addition doesn't need to be performed. For example, equation 4 can be decomposed into x and y components as follows:

	<math>x_f = \frac{1}{2} a_x \cdot t^2 + v_{x, i} \cdot t + x_i</math>		<math>x_f = \frac{1}{2} a_x \cdot t^2 + v_{x, i} \cdot t + x_i</math>

Richard.Udall: /* Further reading */

2019-07-25T19:59:40Z

Richard.Udall at 19:59, 24 July 2019

2019-07-24T19:59:21Z

← Older revision		Revision as of 15:59, 24 July 2019
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	'''Edited by Richard Udall July 2019'''		'''Edited by Richard Udall July 2019'''


	Kinematics is the branch of mechanics which describes the motions of particles, using knowledge only of position, time, and their related quantities velocity and acceleration (formally, these are the derivatives of position with respect to time). As such, kinematics is used once one already knows all the forces acting on the particle, at which point one uses Newton's second law to determine acceleration and proceed from there. The kinematics discussed here will specifically be constant acceleration kinematics, meaning that <math> \vec{a} </math> will be a vector with fixed magnitude and direction for the duration of the given problem. This therefore constitutes a small and heavily idealized set of conditions, but they are instructive nonetheless.		Kinematics is the branch of mechanics which describes the motions of particles, using knowledge only of position, time, and their related quantities velocity and acceleration (formally, these are the derivatives of position with respect to time). As such, kinematics is used once one already knows all the forces acting on the particle, at which point one uses Newton's second law to determine acceleration and proceed from there. The kinematics discussed here will specifically be constant acceleration kinematics, meaning that <math> \vec{a} </math> will be a vector with fixed magnitude and direction for the duration of the given problem. This therefore constitutes a small and heavily idealized set of conditions, but they are instructive nonetheless.

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	*[[Projectile Motion]]		*[[Projectile Motion]]
	*[[Newton's Second Law: the Momentum Principle]]		*[[Newton's Second Law: the Momentum Principle]]
			*[[Derivation of Average Velocity]]

	===Further reading===		===Further reading===

← Older revision		Revision as of 15:59, 25 July 2019
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	===Further reading===		===Further reading===
	Matter and Interactions, 4th Edition
			*Matter and Interactions, 4th Edition

	===External Links===		===External Links===