Uniform acceleration

Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period.

Free fall

The most frequently cited example of uniform acceleration however is that of an object in free fall. The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength g (also called acceleration due to gravity). 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 objects of different masses, where resistances to motion can be neglected, accelerate at the same rate. It is commonly stated that this was first demonstrated by Galileo Galilei, although many historians state that this did not in fact take place.

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.

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.

See also

• Kinematics
• Equations of motion

Further reading

• L. Marder (1957). On uniform acceleration in special and general relativity. Mathematical Proceedings of the Cambridge Philosophical Society, 53, pp 194-198 doi:10.1017/S0305004100032114