drive the equation bof motion x = ut + ½at² vel
Answers
Correct Question :-
- Drive the equation of motion s = ut + ½ at²
To prove :-
- s = ut + ½ at²
Proof :-
As we know that
- Average velocity = u + v/2
Where " u " is initial velocity and " v " is final velocity.
Distance travelled = Average velocity × time
→ s = (u + v)/2 × t
According to first equation of motion
→ v = u + at
So, put the value of " v "
→ s = (u + u + at)/2 × t
→ s = (2u + at)/2 × t
→ s = 2ut/2 + at²/2
→ s = ut + ½ at²
- Hence proved
Derive the second equation of motion :-
Second equation of motion
Let us consider, an object is moving along a straight line with a uniform acceleration (a) at t = 0 , initial velocity (u) . Let s be the distance travelled by the object in time t , from A to b.
↠ s = area enclosed by velocity time graph and time axis.
↠ s = Area of OABC
↠ s = Area of OACD + Area of ∆ ABD.
↠ s = OA × AD + 1/2 × BD × AD
↠ s = OA × OC + 1/2 (BC - CD) × OC. ...Eq(1)
OA = u
OC = t
BD = BC - CD = v - u.
↠ s = u × t + 1/2 (v - u) t
From first equation of motion we have , v - u = at.
↠ s = u × t × 1/2 × at × t
Hence ,
Derived !
Required information :-
You can derive first equation also in the same way .
✮ First Equation of Motion
Let us consider an object is moving with uniform acceleration along a straight line . Let (u) be the initial velocity at t = 0 . After interval of time (t) , its velocity becomes u.
↠ Let OA = u [ t = 0 ]
Here in the figure we draw AD ⟂ BC , and BE ⟂ OY.
↠ ∠BAD = ϴ. [ say ]
Now,
↠ Acceleration = slope of Line AB
↠ a = tan ϴ
↠ a = BD/AD. [ tan ϴ = P/B ]
↠ a = (v - u)/t. [ AD = OC = t ]
Or
More related information :-
✮ First equation of motion
❥ v = u + at.
v = final velocity.
u = initial velocity.
a = acceleration.
t = time taken.
Note :
first equation of motion also called velocity time equation.
✮ Second equation of motion
↠ s = ut + ½ at².
u = initial velocity.
a = acceleration.
t = time taken.
s = distance covered.
Note :-
Second equation of motion also called positive time equation.
✮ Third equation of motion
❥ v² - u² = 2as
v = final velocity.
u = initial velocity.
a = acceleration.
Note :-
Third equation of motion also called velocity positive equation.
❥ First law of motion
↠ first law of motion states that an object continues to be in a state of rest or of uniform motion along a straight line unless acted upon by an unbalanced force.
❥ Second law of motion
↠ second law of motion states that the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the force.
❥ Third law of motion
↠ third law of motion states that for every action there is an equal and opposite reaction and they act on two different bodies.