Question

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Drive the second equation of motion is - S=ut +1 at
2​

Answer 1

Derive the second equation of motion s = ut + (1/2)at^2 using the velocity-time graph. Derive the second equation of motion s = ut + (1/2)at2 using the velocity-time graph. Consider the linear motion of a body with an initial velocity u. Let the body accelerate uniformly and acquire a final velocity v after time t.

Answer 2

★ Correct Question :-

• Derive the second equation of motion :-

Answer :-

• 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

$\therefore\underline{\boxed{\textsf{s = {\textbf{ut + 1/2at²}}}}} \qquad\qquad \bigg\lgroup\bold{2nd \ equation \ of \ motion} \bigg\rgroup$

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 ]

$\therefore\underline{\boxed{\textsf{v = {\textbf{u + at}}}}} \qquad\qquad \bigg\lgroup\bold{1st \ equation \ of \ motion} \bigg\rgroup$

Or

$\underline{\boxed{\textsf{at = {\textbf{v - u}}}}} \qquad\qquad \bigg\lgroup\bold{1st \ equation \ of \ motion} \bigg\rgroup$

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.

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