Acceleration - Revision history

Estone41 at 04:36, 5 December 2021

2021-12-05T04:36:31Z

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 This page defines and describes acceleration.
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 [[Iterative Prediction]] is used to update the position of the particle. Click "view this program" in the top left corner to view the source code.
 ==Examples==

YorickAndeweg: /* A 1-Dimensional Graphical Model */

2019-08-08T18:10:32Z

A 1-Dimensional Graphical Model

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	<li>The slope of a position-vs-time graph at any point in time is the velocity of the particle at that point in time.</li>		<li>The slope of a position-vs-time graph at any point in time is the velocity of the particle at that point in time.</li>
	<li>The slope of a velocity-vs-time graph at any point in time is the acceleration of the particle at that point in time.</li>		<li>The slope of a velocity-vs-time graph at any point in time is the acceleration of the particle at that point in time.</li>
	<li>The area under a velocity-vs-time graph ~~up to any point in~~ time is the position of the particle at that ~~point in~~ time.</li>		<li>The area under a velocity-vs-time graph over a time interval is the change in the position of the particle over that time period.</li>
	<li>The area under an acceleration-vs-time graph ~~up to any point in~~ time is the velocity of the particle at that ~~point in~~ time.</li>		<li>The area under an acceleration-vs-time graph over a time interval is the change in the velocity of the particle over that time period.</li>
	</ol>		</ol>

YorickAndeweg: /* 5. (Middling) */

2019-08-01T16:53:06Z

5. (Middling)

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	Question:		Question:

	A particle is travelling at a speed of 5m/s. The particle undergoes a nonconstant acceleration in the direction of its velocity for 1 second. The magnitude of the acceleration is given by <math> a = 6 - 6t^2</math>, where t=0 is the beginning of the 1 second interval. What is the final ~~velocity~~ of the particle?		A particle is travelling at a speed of 5m/s. The particle undergoes a nonconstant acceleration in the direction of its velocity for 1 second. The magnitude of the acceleration is given by <math> a = 6 - 6t^2</math>, where t=0 is the beginning of the 1 second interval. What is the final speed of the particle?

YorickAndeweg: /* A Mathematical Model */

2019-08-01T16:51:24Z

A Mathematical Model

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	where <math>\vec{v}</math> is a velocity vector and <math>t</math> is time.		where <math>\vec{v}</math> is a velocity vector and <math>t</math> is time.

			====Effect on Velocity Vector====

			Acceleration indicates a change in the velocity vector's magnitude, direction, or both:
			#Any component of the acceleration vector parallel to the velocity vector indicates an increase in velocity magnitude/ speed.
			#Any component of the acceleration vector anti-parallel to the velocity vector indicates a decrease in velocity magnitude/ speed.
			#Any component of the acceleration vector perpendicular to the velocity vector indicates a change in the velocity vector's direction.

	====Average Acceleration====		====Average Acceleration====

YorickAndeweg: /* A Graphical Model */

2019-08-01T16:47:25Z

A Graphical Model

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	This makes sense because acceleration should be directly proportional to force, which causes it, and inversely proportional to mass, which resists it (see [[Inertia]]). This equation is often used to find the acceleration of a particle. If that acceleration is constant, kinematic equations are then often used to determine how its position and velocity change over time.		This makes sense because acceleration should be directly proportional to force, which causes it, and inversely proportional to mass, which resists it (see [[Inertia]]). This equation is often used to find the acceleration of a particle. If that acceleration is constant, kinematic equations are then often used to determine how its position and velocity change over time.

	===A Graphical Model===		===A 1-Dimensional Graphical Model===

	Below are some graphs relating velocity and position along one dimension to time as a result of varying types of acceleration. Because of the derivative and integral relationships between position, velocity, and acceleration:		Below are some graphs relating velocity and position along one dimension to time as a result of varying types of acceleration. Because of the derivative and integral relationships between position, velocity, and acceleration:

YorickAndeweg: /* The Main Idea */

2019-08-01T16:46:57Z

The Main Idea

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	==The Main Idea==		==The Main Idea==

	Acceleration, denoted by the symbol <math>\vec{a}</math>, is a vector quantity defined as the rate of change of [[Velocity]] with respect to time. In calculus terms, it is the time derivative of the velocity vector. Acceleration ~~may indicate~~ a change in the velocity vector's magnitude, direction, or both. Acceleration is an instantaneous value, so it may change over time. In physics, the acceleration of particles is caused by forces. The most commonly used metric unit for acceleration is the meter per second per second (m/s/s or m/s<sup>2</sup>).		Acceleration, denoted by the symbol <math>\vec{a}</math>, is a vector quantity defined as the rate of change of [[Velocity]] with respect to time. In calculus terms, it is the time derivative of the velocity vector. Acceleration indicates a change in the velocity vector's magnitude, direction, or both. Acceleration is an instantaneous value, so it may change over time. In physics, the acceleration of particles is caused by forces. The most commonly used metric unit for acceleration is the meter per second per second (m/s/s or m/s<sup>2</sup>).

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

YorickAndeweg: /* The Main Idea */

2019-08-01T16:46:32Z

The Main Idea

← Older revision		Revision as of 12:46, 1 August 2019
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	==The Main Idea==		==The Main Idea==

	Acceleration, denoted by the symbol <math>\vec{a}</math>, is a vector quantity defined as the rate of change of ~~velocity~~ with respect to time. In calculus terms, it is the time derivative of the velocity vector. Acceleration is an instantaneous value, so it may change over time. In physics, the acceleration of particles is caused by forces. The most commonly used metric unit for acceleration is the meter per second per second (m/s/s or m/s<sup>2</sup>).		Acceleration, denoted by the symbol <math>\vec{a}</math>, is a vector quantity defined as the rate of change of [[Velocity]] with respect to time. In calculus terms, it is the time derivative of the velocity vector. Acceleration may indicate a change in the velocity vector's magnitude, direction, or both. Acceleration is an instantaneous value, so it may change over time. In physics, the acceleration of particles is caused by forces. The most commonly used metric unit for acceleration is the meter per second per second (m/s/s or m/s<sup>2</sup>).

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

YorickAndeweg: /* Average Acceleration */

2019-08-01T16:43:57Z

Average Acceleration

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	<math>\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}</math>.		<math>\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}</math>.

	As the duration of the time interval <math>\~~Deltat~~ t</math> becomes very small, the value of the average acceleration over that time interval approaches the value of the instantaneous acceleration at any point in time within that interval.		As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average acceleration over that time interval approaches the value of the instantaneous acceleration at any point in time within that interval.

	====Derivative Relationships====		====Derivative Relationships====

YorickAndeweg: /* Average Acceleration */

2019-08-01T16:42:28Z

Average Acceleration

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	<math>\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}</math>.		<math>\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}</math>.

			As the duration of the time interval <math>\Deltat t</math> becomes very small, the value of the average acceleration over that time interval approaches the value of the instantaneous acceleration at any point in time within that interval.

	====Derivative Relationships====		====Derivative Relationships====

YorickAndeweg: /* A Computational Model */

2019-08-01T16:28:32Z

A Computational Model

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	In the simulation below, a particle initially moving to the right undergoes a leftward acceleration followed by an upward acceleration. The purple arrow represents the particle's velocity and the red arrow represents its acceleration. Notice that the tip of the purple arrow always moves in the direction indicated by the red arrow.		In the simulation below, a particle initially moving to the right undergoes a leftward acceleration followed by an upward acceleration. The purple arrow represents the particle's velocity and the red arrow represents its acceleration. Notice that the tip of the purple arrow always moves in the direction indicated by the red arrow.

	[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Acceleration Click here for acceleration simulation]]		[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Acceleration Click here for acceleration simulation]

	[[Iterative Prediction]] is used to update the position of the particle. Click "view this program" in the top left corner to view the source code.		[[Iterative Prediction]] is used to update the position of the particle. Click "view this program" in the top left corner to view the source code.