There are three basic equations of motion for bodies moving with uniform acceleration. These Equations are used to calculate the displacement (s), velocity(v), Time (t), and acceleration (a) of a moving body.
Suppose a body is moving with uniform acceleration “a” during some time interval “t”. Its initial velocity “vi” changes and is denoted as final velocity “vf”. If the body covers a distance “S” in this duration of time.
In the First equation determine the final velocity of a uniformly accelerated body. Where,
\(\vec{vi}\) = initial velocity
\(\vec{vf}\)= final velocity
\(\vec{a}\)= acceleration
t = time
The average acceleration is the change in velocity over a time interval
acceleration = changes in velocity /time taken
\(\vec{a}\)=\(\frac{(\vec{v}_f – \vec{v}_i)}{t}\)
\(\vec{a}\)t = \(\vec{vf}\)– \(\vec{vi}\)
\(\vec{a}\)t + \(\vec{vi}\)= \(\vec{vf}\)
\(\vec{vf}\)= \(\vec{vi}\) + \(\vec{a}\)t
The second equation of motion determines the distance covered during some time interval “t”, while a body is accelerating from a known initial velocity.
As we know, average velocity:
\(v_{avg} = \frac{v_f + v_i}{2}\)
\(v_{avg}=\frac{v_i + at + v_i}{2}\)
\(v_{avg}= \frac{2v_i + at}{2}\)
\(v_{avg}=\frac{2v_i}{2} + \frac{at}{2}\)
\(v_{avg}=v_i + \frac{1}{2}at\)
Velocity can be given as:
\(v_{avg}=\frac{s}{t}\)
\(\frac{s}{t}=v_i + \frac{1}{2}at\)
\(S=(v_i + \frac{1}{2}at)t\)
\(S = v_i t + \frac{1}{2}at^2\)
Third equation of motion determines the relationship among the velocity and the distance covered by a uniformly accelerated body, where the time interval is not mentioned.
Let us take the first equation of motion.
\(\vec{vf}\)= \(\vec{vi}\) + \(\vec{a}\)t
By squaring both sides of the equation we get:
\(v_f^2 = (v_i + at)^2\)
We know
\( (a+b)^2 = a^2 + 2ab + b^2 \)
\( v_f^2 = v_i^2 + 2v_iat + a^2t^2 \)
\( v_f^2 = v_i^2 + 2a(v_i t + \frac{1}{2}at^2) \)
By second equation of motion
\(S = v_i t + \frac{1}{2}at^2\)
\( v_f^2 = v_i^2 + 2aS \)
\( 2aS = v_f^2 – v_i^2 \)
For the motion of bodies under the influence of gravity the equations of motion are slightly modified where distance is taken as (S=h) and acceleration is taken as g (a=g).
Therefore the equation of motion is taken as.
\(\vec{vf}\)= \(\vec{vi}\) + \(\vec{a}\)t
\(S = v_i t + \frac{1}{2}at^2\)
\( 2aS = v_f^2 – v_i^2 \)
