Acceleration: Difference between revisions

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'''Solution:'''
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:<math> \vec{a}(t) = \frac{d \vec{v}}{dt} </math>
:<math> \vec{a}(t) = \frac{d \vec{v}(t)}{dt} </math>


:<math> \vec{a}(t) = \frac{d}{dt} <6t, 10 \sin t, 4> </math>
:<math> \vec{a}(t) = \frac{d}{dt} <6t, 10 \sin t, 4> </math>

Revision as of 20:31, 6 June 2019

This page defines and describes acceleration.

The Main Idea

Acceleration, denoted by the symbol [math]\displaystyle{ \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 velocity. In physics, the acceleration of particles is caused by forces. Acceleration is an instantaneous value, so it may change over time. The most commonly used metric unit for acceleration is the meter per second per second (m/s/s or m/s2).

A Mathematical Model

Instantaneous acceleration is defined as:

[math]\displaystyle{ \vec{a} = \frac{d\vec{v}}{d t} }[/math].

Average Acceleration

Since acceleration is an instantaneous quantity, it can change over time. Over any given time interval, there is an average acceleration value denoted [math]\displaystyle{ \vec{a}_{avg} }[/math]. The average acceleration over an interval of time [math]\displaystyle{ \Delta t }[/math] is given by

[math]\displaystyle{ \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} }[/math].

Derivative Relationships

Acceleration is the time derivative of velocity, which is in turn the time derivative of position, so acceleration is the second time derivative of position.

Recall that

[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} }[/math]

and

[math]\displaystyle{ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} }[/math].

Therefore:

[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2} }[/math].

Integral Relationships

The integral of acceleration yields velocity, and the integral of velocity yields position, so the double integral of acceleration yields position.

[math]\displaystyle{ \vec{v}(t) = \int \vec{a}(t) \ dt }[/math]

and

[math]\displaystyle{ \vec{r}(t) = \int \vec{v}(t) \ dt }[/math].

Therefore

[math]\displaystyle{ \vec{r}(t) = \iint \vec{a}(t) \ dt \ dt }[/math].

Kinematic Equations

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the Kinematics page.

Newton's Second Law of Motion

In physics, the acceleration of particles is caused by forces. Newton's Second Law: the Momentum Principle states that

[math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math].

From this relation, it can be derived that

[math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math]

, or that

[math]\displaystyle{ \vec{a} = \frac{\vec{F}_{net}}{m} }[/math].

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

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:

  1. The slope of a position-vs-time graph at any point in time is the velocity of the particle at that point in time.
  2. The slope of a velocity-vs-time graph at any point in time is the acceleration of the particle at that point in time.
  3. 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.
  4. 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.

Constant Acceleration

When the acceleration of a particle is constant over time, its velocity is changing at a constant rate, and therefore is graphed as a straight line. The acceleration is the slope of the velocity graph; a positive acceleration means the velocity is increasing and should have a positive slope, and a negative acceleration means the velocity is decreasing and has a negative slope.

If the velocity of a particle is constant over time, the slope of the graph and the acceleration of the particle are 0.

Note that in both of the above graphs, the y-intercept was at a velocity of 0. This is because it was assumed that the particles began at rest. If a particle has other initial conditions, the graph must reflect them. For example, below is the graph of the velocity of a vertically thrown ball over time (see Projectile Motion).

Its initial velocity is positive because the ball is thrown upwards, which we define as the positive direction. However, as time goes on, its velocity decreases at a steady rate due to the constant acceleration of gravity, until it reaches 0. At this point, the ball has reached the highest point in its trajectory; it is moving neither upwards nor downwards. Gravity continues to accelerate the ball downwards, so its velocity becomes negative and the ball moves back towards earth. If the ball lands at the same height as that from which it was thrown, the positive area under the left side of the graph equals the negative area under the right side of the graph because the distance traveled upwards during the first half of the flight equals the distance traveled downwards in the second half of the flight. The total area under the graph would be 0 because the final position is the same as the initial position. The slope of the velocity graph is -g at all points of this graph.

In addition to velocity, it can be useful to visualize the position of a particle over time when it undergoes constant acceleration. The graph below shoes the position of a particle over time when it begins at rest at the origin and undergoes constant positive acceleration.

At first, the position of the particle is barely changing over time, because in the beginning of the particle's motion, its velocity is still small. However, as the velocity steadily increases as a result of the constant positive acceleration, the slope of the position graph increases, showing that towards the end of its motion, the particle is moving greater distances in shorter time intervals. The slope of the position graph, which is the velocity of the particle, is increasing at a steady rate, resulting in a parabolic position graph. The kinematic equation corresponding to this graph is [math]\displaystyle{ r_f = \frac{1}{2} a * t^2 + v_i * t + r_i }[/math].

Nonconstant Acceleration

When acceleration is not constant, the velocity-vs-time graph is not a straight line but a curve. The nature of the curve depends on how the acceleration changes. An everyday example of an object that accelerates non-uniformly is a car. Cars typically undergo positive but decreasing acceleration when they begin moving from rest. This is what a car's velocity-vs-time graph might look like:

At first, the car has a high acceleration, and its velocity changes rapidly. However, as it approaches its target speed, its acceleration decreases.

Examples

Some of these example problems make use of the kinematic equations. For more practice using those, try the practice problems on the Kinematics page.

1. (Simple)

Question:

A motorcycle accelerates uniformly from rest to a speed of 7.10 m/s over a distance of 35.4 m. Determine the acceleration of the motorcycle.


Solution:


Given:

[math]\displaystyle{ v_i = 0 m/s }[/math]
[math]\displaystyle{ v_f = 7.10 m/s }[/math]
[math]\displaystyle{ d = 35.4 m }[/math]

Find:

[math]\displaystyle{ a = ? }[/math]

Calculations:

[math]\displaystyle{ v_f^2 = v_i^2 + 2ad }[/math]
[math]\displaystyle{ 2ad = v_f^2 - v_i^2 }[/math]
[math]\displaystyle{ a = \frac{v_f^2 - v_i^2}{2*d} }[/math]
[math]\displaystyle{ a = \frac{(7.10m/s)^2 - (0m/s)^2}{2*35.4m} }[/math]
[math]\displaystyle{ a = 0.712 m/s^2 }[/math]

2. (Simple)

Question:

A rabbit can jump to a height of 2.62 m. Determine the takeoff speed of the rabbit.


Solution:

Given:

[math]\displaystyle{ a = -9.8 m/s^2 }[/math]
[math]\displaystyle{ v_f = 0 m/s }[/math] *Considering the moment of takeoff to maximum height
[math]\displaystyle{ d = 2.62 m }[/math]

Find:

[math]\displaystyle{ v_i = ? }[/math]

Calculations:

[math]\displaystyle{ v_f^2 = v_i^2 + 2ad }[/math]
[math]\displaystyle{ v_i^2 = 2ad - v_f^2 }[/math]
[math]\displaystyle{ v_i = \sqrt{2ad - v_f^2} }[/math]
[math]\displaystyle{ v_i = \sqrt{2*-9.8m/s^2*2.62m - (0m/s)^2} }[/math]
[math]\displaystyle{ v_i = 7.17 m/s }[/math]

3. (simple)

Question:

A rock is dropped from a bridge into a lake and is heard to hit the water 3.41 s after being dropped. Determine the height of the bridge above the water.


Solution:

Given:

[math]\displaystyle{ a = -9.8 m/s^2 }[/math]
[math]\displaystyle{ t = 3.41 s }[/math]
[math]\displaystyle{ v_i = 0 m/s }[/math]

Find:

[math]\displaystyle{ d = ? }[/math]

Calculations:

[math]\displaystyle{ d = v_i*t + 0.5*a*t^2 }[/math]
[math]\displaystyle{ d = (0 m/s)*(3.41 s)+ 0.5*(-9.8 m/s^2)*(3.41 s)^2 }[/math]
[math]\displaystyle{ d = 0 m + 0.5*(-9.8 m/s^2)*(11.63 s^2) }[/math]
[math]\displaystyle{ d = -57.0 m }[/math] (57.0 meters deep)

4. (Simple)

Question:

The initial velocity of a particle is <-4, 3, -12>m/s. After 4 seconds, the velocity of the particle is <-8, 3, 8>m/s. What was the particle's average acceleration during the 4 second interval?


Solution:

[math]\displaystyle{ \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t} = \frac{\lt -8, 3, 8\gt - \lt -4, 3, -12\gt }{4} = \frac{\lt -4, 0, 20\gt }{4} = \lt -1, 0, 5\gt }[/math]m/s

Note that at any given moment, the acceleration of the particle may be something other than <-1, 0, 5>; this is simply the average acceleration over the 4s interval. (However, according to the mean value theorem, there must be some point in time within the interval at which the acceleration of the particle is the average acceleration.)

5. (Middling)

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]\displaystyle{ a = 6 - 6t^2 }[/math], where t=0 is the beginning of the 1 second interval. What is the final velocity of the particle?


Solution:

[math]\displaystyle{ v_f = v_i + \int_{t_i}^{t_f} a(t) \ dt }[/math]
[math]\displaystyle{ v_f = 5 + \int_0^1 6 - 6t^2 \ dt }[/math]
[math]\displaystyle{ v_f = 5 + [6t - 2t^3]_0^1 }[/math]
[math]\displaystyle{ v_f = 5 + 4 = 9 }[/math]m/s

6. (Middling)

Question:

The velocity of a particle as a function of time is given by the expression [math]\displaystyle{ \vec{v}(t) = \lt 6t, 10 \sin t, 4\gt }[/math]. What is the acceleration of the particle as a function of time?


Solution:

[math]\displaystyle{ \vec{a}(t) = \frac{d \vec{v}(t)}{dt} }[/math]
[math]\displaystyle{ \vec{a}(t) = \frac{d}{dt} \lt 6t, 10 \sin t, 4\gt }[/math]
[math]\displaystyle{ \vec{a}(t) = \lt 6, 10 \cos t, 0\gt }[/math]

Connectedness

  1. Acceleration can be applied to many things such as calculating free fall and predicting motion, time and force.
  2. Acceleration is applied almost everywhere. For example, in biology, it can be used to calculate the growth of plants over time, to predict motion of animals, and even to calculate winds to predict climate changes.
  3. An interesting industrial application of acceleration is to calculate impacts on cars and to improve quality of manufacturing car safety.


See also


External links

  • A physics resource written by experts for an expert audience Physics Portal
  • A wiki book on modern physics Modern Physics Wiki
  • The MIT open courseware for intro physics MITOCW Wiki
  • An online concept map of intro physics HyperPhysics
  • Interactive physics simulations PhET
  • OpenStax algebra based intro physics textbook College Physics
  • The Open Source Physics project is a collection of online physics resources OSP
  • A resource guide compiled by the AAPT for educators ComPADRE


References

  • Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Hoboken, NJ: John Wiley & Sons, 2015. Print.







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