Uniform acceleration

Uniform, or constant, acceleration is a type of motion in which the velocity of an object changes equal amounts in equal time periods. An example of an object having uniform acceleration would be a ball rolling down a ramp. The object picks up velocity as it goes down the ramp with equal changes in time. The most frequently cited example of uniform acceleration however is that of an object in free fall. Here it can be seen that the acceleration of a falling body in the absence of resistances to motion (friction etc.) is dependent only on the gravitational field strength g (also called acceleration due to gravity), since by Newton's Second Law the force, F, acting on a body is given by:

$\mathbf {F} =m\mathbf {g}$

and similarly the acceleration, a, of a body is given by:

$\mathbf {a} ={\mathbf {F} \over {m}}$

where m is the object's mass in each case. By combining these two expressions it can be seen that:

$\mathbf {a} =\mathbf {g}$

The consequence of this is that, as demonstrated by Galileo Galilei, objects of different masses, where resistances to motion can be neglected, accelerate at the same rate - a hammer and a feather, released from the same height in a vacuum, hit the ground at the same time.

Circular motion

A further important example of a body experiencing uniform acceleration is one which is in uniform horizontal circular motion. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's velocity also changes. A change in velocity over time is an acceleration. This acceleration is directed toward the centre of the circle and takes the value:

$a={{v^{2}} \over {r}}$

where v is the object's speed. Equivalently, the radial acceleration may be calculated from the object's angular velocity $\omega$, whence:

$\mathbf {a} ={-\omega ^{2}}\mathbf {r} .$

It is important to note that the acceleration, hence also the force, acting on a body in uniform horizontal circular motion is directed toward the centre of the circle, i.e. it is centripetal - the so called 'centrifugal force' appearing to act outward on a body is really a pseudo force experienced due to the body's linear momentum at a tangent to the circle.

Formulae

Due to the unique algebraic properties of constant acceleration, mathematicians have derived a number of formulae which may be used to determine any of the following quantities: displacement, initial velocity, final velocity, acceleration and time.

These are as follows:

$\mathbf {v} =\mathbf {u} +\mathbf {a} t$

$v^{2}=u^{2}+2\mathbf {a} \cdot \mathbf {s}$

$\mathbf {s} =\mathbf {u} t+{{1} \over {2}}\mathbf {a} t^{2}$

$\mathbf {s} ={{(\mathbf {u} +\mathbf {v} )t} \over {2}}$

where

$\mathbf {s}$ = displacement

$\mathbf {u}$ = initial velocity

$\mathbf {v}$ = final velocity

$\mathbf {a}$ = uniform acceleration

$t$ = time.

See also

• Kinematics
• Equations of Motion