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constant acceleration

Three simple equations of motion may be used when acceleration is constant in both magnitude and direction, or when acceleration is constant in magnitude and the motion is forced to follow a fixed track.

These three equations apply only in the direction of the acceleration: in perpendicular directions, the component of acceleration is zero, and so the component of displacement (distance) will be a constant times time.

Each equation involves only four of the five variables [itex]a\ u\ v\ s\ \text{and}\ t[/itex] (other symbols used are [itex]u_i\ \text{and}\ u_f[/itex] for [itex]u\ \text{and}\ v[/itex], and [itex]\Delta s\ \text{and}\ \Delta t[/itex] for [itex]s\ \text{and}\ t[/itex]), so choose whichever equation omits the variable you are not interested in.

[itex]s[/itex] in these equations is the displacement (distance) relative to a stationary point. If the displacement given is between two bodies of which one has constant acceleration and the other has constant velocity, the equations may be applied in a frame in which the latter is stationary.

In the direction of constant acceleration:

[tex]v\ =\ u\ +\ at[/tex]

[tex]v^2\ =\ u^2\ +\ 2as[/tex]

[tex]s\ =\ ut\ +\ \frac{1}{2}at^2[/tex]

Perpendicular to the direction of constant acceleration:

[tex]v\ =\ u[/tex]

[tex]s\ =\ ut[/tex]

For circular motion, with angular displacement [itex]\theta[/itex], angular velocity [itex]\omega[/itex], and angular acceleration [itex]\alpha[/itex]:

[tex]\omega_f\ =\ \omega_i\ +\ \alpha t[/tex]

[tex]\omega_f ^2\ =\ \omega_i^2\ +\ 2\alpha\theta[/tex]

[tex]\theta\ =\ \omega_it\ +\ \frac{1}{2}\alpha t^2[/tex]


Recent forum threads on constant acceleration
How constant acceleration affects the tripulation of a spaceship?
Kinematics of Constant acceleration word problem
Non constant acceleration in gravitation field
Vi for Constant Acceleration Kinematics
Trouble Deriving a Constant Acceleration Formula
> Classical Mechanics
>> Newtonian Dynamics

See Also


Extended explanation
Motion of a projectile:

A ball thrown at speed [itex]V[/itex] at an angle [itex]\theta[/itex] above the horizontal will, using the usual [itex](x,y)[/itex] coordinates, have acceleration [itex](0,-g)[/itex], and horizontal and vertical components of displacement:
[itex]x\ =\ (V\cos\theta )t[/itex]

[itex]y\ =\ (V\sin\theta )t\ -\ \frac{1}{2}gt^2[/itex]
Why two solutions?

Exam questions often ask you to find the speed angle or time for a ball to reach its target.

There are usually two correct solutions, one is a "drive" and the other is a "lob", but they both take the same time, t, to get to the same place, (x,y).

They result from simultaneously solving the above equations, giving a quadratic equation in t.

Relative motion:

If body A has constant acceleration, and body B has constant velocity, and if the initial distance between them is given (or sought), then change to a frame of reference in which B is stationary.

The acceleration in that frame will be the same, but A's velocity will be replaced by its velocity relative to B.

In that frame, the given distance is now measured relative to a stationary point, and may be used as [itex]s[/itex].

Motion along a curved track:

A vehicle constantly accelerating or braking along a curved track obeys the same three equations.

Circular motion:

For an object moving in a circle, the same three equations may be used with angular acceleration velocity and displacement instead of ordinary (linear) acceleration velocity and displacement.


tiny-tim @ 07:03 AM Sep3-11
Added "drive" and "lob", since some threads have been asking how can there be two solutions

tiny-tim @ 05:43 PM Feb9-09
on a forum search, "constant acceleration" got 500 hits in 15 months, "kinematic equations" got 500 hits in 42 months, and "kinematic equation" got 326 hits

Kurdt @ 04:44 PM Feb9-09
I think "kinematic equations" comes up more often than "constant acceleration". I certainly use it more frequently.

tiny-tim @ 09:44 AM Feb7-09
Thanks, Redbelly! And I think you're right about motion of a projectile … I've changed it to V.
Definitions? I know you like putting them all in , but I prefer a minimalist approach … so long as the notation is a recognised standard.
I preferred to leave out the average speed equation (the fourth musketeer?), partly because (as you say) it rarely comes up, and partly because I reckoned it easier and safer to work it out each time it's needed than to learn it.

EDIT by Redbelly98:
That's fine, they were just suggestions. And the average velocity equation is now listed in the Commentary, so I'm happy with that. I remember it by thinking of two expressions for average velocity, and equating them ... no need to memorize it or spend time deriving it.

Redbelly98 @ 08:23 AM Feb7-09
I think the title "constant acceleration" is appropriate. Remember that autolinking happens only for exact matches to the title. If it's too descriptive and wordy, we'd seldom have autolinks where they would be helpful.

cristo @ 05:32 PM Feb5-09
I think the title should be changed to something more descriptive.

Redbelly98 @ 08:57 AM Feb3-09
Looks good t-t! I have 2 comments:

1. There is a 4th equation involving 4 of the 5 variables:

(v+u)/2 = s/t

This is essentially a statement about average velocity. It would be used in problems where the acceleration is neither given nor asked for, though this happens rarely.

2. How about including definitions of all terms, in particular u and v? Initial velocity seems to be u, but in the Motion of a projectile subsection v is used for initial speed.

Thanks for doing this, hard to believe it wasn't done earlier!

tiny-tim @ 02:45 PM Feb2-09
Created because a surprising number of PF members seem to know only one or two of the three equations.