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Linear Velocity and Acceleration

Introduction to linear velocity and acceleration with an emphasis on constant acceleration.


Key facts

Velocity is the rate of change of position with respect to time:

v = \frac{ds}{dt}

Acceleration is the rate of change of velocity with respect to time:

a = \frac{dv}{dt} (=\dot v)
a = \frac{d^2s}{dt^2} (=\ddot s)
a = v \frac{dv}{ds}

If \inline u is the initial velocity, \inline v the final velocity, and \inline s the distance traveled in a time \inline t, then:

v = u + at
s = ut + \frac{1}{2} at^2

Velocity and acceleration are vector quantities both measured with respect to time, where acceleration is the rate at which the object changes its velocity. Constant acceleration, the term in consideration here, assumes that the rate of change of velocity over a period of time is constant. In other words, if you were in a bus that was in a state of constant acceleration, we would say that the bus was either speeding up or slowing down at a steady, constant rate. From this approximation, the position, speed and final velocity due to the acceleration can be calculated.

Linear Velocity

In order to define linear velocity, consider a particle moving in a straight line from a fixed point \inline O, as diagramed in Figure 1.

Figure 1

At a time \inline t let the distance from \inline O be \inline s, and at a time \inline t+\delta\,t let the distance increase to \inline s+\delta\,s. So over a time of \inline \delta\,t the particle traveled the distance \inline \delta\,s, from which it can be seen that the average rate of displacement from \inline O, or average velocity, is \inline \displaystyle \frac{\delta\,s}{\delta\,t}. The velocity at time \inline t is defined as the limiting value of this quantity as \inline \delta\,t\rightarrow 0:

Thus, velocity can be defined as the rate of change of position with respect to time.

Similarly, it can be shown that acceleration, which is the rate of change of velocity with respect to time, is \inline \displaystyle\frac{dv}{dt}, or, expressed in a different form, \inline \displaystyle\frac{d^2s}{dt^2}. Acceleration can also be written in a third form which is independent of time. For example, by expressing the acceleration as:

\frac{dv}{dt} = \frac{dv}{ds} \frac{ds}{dt}

and taking into account (1), we get:

\frac{dv}{dt} = v \frac{dv}{ds}

Hence, acceleration may be expressed in any of the following three forms:

\frac{dv}{dt};\ \frac{d^2s}{dt^2};\ v\:\frac{dv}{ds}

It is usual in mechanics to denote differential coefficients with respect to time by dots placed above the dependent variables. Therefore, notations such as \inline \displaystyle \frac{dv}{dt}, \inline \displaystyle \frac{ds}{dt}, \inline \displaystyle \frac{d^2s}{dt^2}, and \inline \displaystyle\frac{d^2x}{dt^2} can also be denoted by \inline \dot{v}, \inline \dot{s}, \inline \ddot{s}, and \inline \ddot{x} respectively.

Constant Acceleration

Let \inline u be the initial velocity, \inline v the final velocity, \inline a the acceleration (which in this case is a constant), \inline s the distance traveled in time \inline t, and \inline t the time.

We previously saw that the acceleration can be written as \inline \displaystyle \frac{dv}{dt}. As in this case \inline a is constant, by integrating \inline a = \displaystyle \frac{dv}{dt} with respect to \inline t, we get:

In this equation, at \inline t = 0, \inline v becomes \inline u, and so the constant equals \inline u. Thus, (2) becomes:

v = u + at

Integrating again with respect to \inline t, and considering that \inline s = \displaystyle\frac{ds}{dt}, we get:

In this equation, at \inline t = 0, \inline s becomes \inline 0, and thus the constant equals \inline 0. Hence, (3) becomes:

s = ut + \frac{1}{2} a t^2

Integrating this equation with respect to \inline s, and considering that \inline v \displaystyle\frac{dv}{ds} = a, we get:

As at \inline s = 0 \inline v becomes \inline u, the constant becomes \inline \displaystyle \frac{1}{2} u^2. Thus (4) becomes:

v^2 = u^2 + 2as

Example -
The driver of an express train traveling at 60 mph sees, on the same track, 600 ft in front of him, a slow train traveling in the same direction at 20 mph.

What is the least retardation that must be applied to the express to avoid a collision?
For the express train, \inline u = 88ft/s, and \inline v = 29.33ft/s. Taking into account that \inline v^2=u^2+2as, we get:

29.33^2 - 88^2 = 2as

In a time \inline t the slow train will have traveled a distance of \inline 29.33t ft. The express train will have gone further and will have traveled \inline 600+29.33t ft:

Moreover, considering that \inline v=u+at, we get:

Combining (1) and (2), we thus obtain:

29.33^2 - 88^2 = 2a \left[ 600+29.33 \left( \frac{29.33-88}{a} \right) \right]

from which:

a = -2.87 ft/s^2

Further examples of constant acceleration can be seen in reference:1702 "Frictionless Projectiles" .