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 + {(at^{2}) / 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 + {(at^{2}) / 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 = −2u^{2} + v^{2} + u^{2}
⇒ 2as = −u^{2} + v^{2}
⇒ 2as + u^{2} = v^{2}
⇒ v^{2} = u^{2} + 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 (at^{2})
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 = v^{2}−u^{2}
⇒ 2as + u^{2} = v^{2}
⇒ v^{2} = u^{2} + 2as
The above expression gives the relation between position and velocity and is called the third equation of motion.