• Equation of Motion

Relation among velocity, distance, time and acceleration is called equations of motion. There are three equations of motion:

First Equation of Motion: The final velocity (v) of a moving object with uniform acceleration (a) after time t.

Let,

u = Initial velocity           v = Final velocity

t = Time                         a = Acceleration

s = Distance covered in given time

We know that,

Acceleration (a) = Change in velocity / Time taken

⇒ a = (Final velocity - Initial velocity) / Time taken

⇒ a = (v − u) / t

⇒ at = v − u

⇒ at − v= − u

⇒ − v = − u − at

v = u + at               --------------   (i)

This equation is known as first equation of motion.

Second Equation of Motion: Distance covered in time (t) by a moving body.

We know that,

Average velocity = (Initial velocity + Final velocity) / 2

∴ Average velocity = (u + v) / 2     ------------  (ii)

Since,

Distance covered (s) in given time = Average velocity x Time

s = Average velocity x Time    --------------- (iii)

After substituting the value of average velocity from equation (ii) we get

⇒ s = [(u + v) / 2] × t

After substituting the value of ‘v’ from first equation of motion we get,

⇒ s = [(u + u + at) / 2] × t

⇒ s = [(2u + at) / 2] × t

s = ut + {(at2) / 2}                 ----------------- (iv)

The is known the second equation of motion.

Third Equation of Motion: The third equation of motion is derived by substituting the value of time (t) from first equation of motion.

Since,

v = u + at

⇒ v − u = at

⇒ at = v − u

⇒ t = (v − u) / a                      -------------------- (v)

We know that the second equation of motion is, s = ut + {(at2) / 2}.  By substituting the value of t from euqation (v), we get-

s = u[(v − u) / a] + {(a[(v − u) / a]2) / 2}

⇒   2s = 2u[(v − u) / a] + [(v − u)2 / a]

⇒ 2as = 2u (v − u) + (v − u)2

⇒ 2as = −2u2 + v2 + u2

⇒ 2as = −u2 + v2

⇒ 2as + u2 = v2

v2 = u2 + 2as                     ------------------ (vi)

This is called the third equation of motion.

Graphical Method

First equation of Motion: Let an object is moving with uniform acceleration.

Let the initial velocity of the object = u  and the object is moving with uniform acceleration, a.  Let object reaches at point B after time, t and its final velocity becomes, v. Draw a line parallel to x-axis DA from point, D from where object starts moving.  Draw another line BA from point B parallel to y-axis which meets at E at y-axis. Let OE = time, t Now, from the graph,

BE = AB + AE

⇒ v = DC + OD (Since, AB = DC and AE = OD)

⇒ v = DC + u (Since, OD = u)

⇒ v = DC + u             ------------------- (i)

Now, Acceleration (a) = Change in velocity time taken

⇒ a = (v − u) / t

⇒ a = (OC − OD) / t = DC / t

⇒ at = DC                    ----------------- (ii)

By substituting the value of DC from (ii) in (i) we get

v = at + u

v = u + at

Above equation is the relation among initial velocity (u), final velocity (v), acceleration (a) and time (t). It is called first equation of motion.

Second equation of Motion: Distance covered by the object in the given time ‘t’ is given by the area of the trapezium ABDOE.

Let in the given time, t the distance covered by the moving object = s

The area of trapezium, ABDOE = Distance (s) = Area of triangle (ABD) + Area of ADOE

⇒ s = [(1/2) × AB × AD] + (OD × OE)

⇒ s = [(1/2) × DC × AD] + (u + t)                 [Since, AB = DC]

⇒ s = [(1/2) × at × t] + ut

⇒ s = [(1/2) × at × t] + ut                               [Since, DC = at]

s = ut + ½ (at2

The above expression gives the distance covered by the object moving with uniform acceleration. This expression is known as second equation of motion.

Third equation of Motion: The distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDOE. Therefore,

Area of trapezium ABDOE = (1/2) × (sum of parallel sides + distance between parallel sides)

⇒ Distance (s) = (1/2) × (DO+BE) × OE

⇒ s = (1/2) × (u + v) × t     ------------ (iii)

Now from equation (ii) a = (v − u) / t

∴ t = (v − u) / a                 ------------- (iv)

After substituting the value of t from equation (iv) in equation (iii)

⇒ s = (1/2) × (u + v) × [(v − u) / a]

⇒ 2as = (v + u) × (v − u)

⇒ 2as = v2−u2

⇒ 2as + u2 = v2

v2 = u2 + 2as

The above expression gives the relation between position and velocity and is called the third equation of motion.