Question

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1.) A body moving with an initial velocity of 1 m/s accelerates uniformly at 0.5 m/s2. Find out its velocity when it has a displacement of 48 m?

2.) For a body moving with an initial velocity u and uniform acceleration a. Find the displacement of the body in time t.

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Answer 1

Answer :-

$\: \\ \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Equations of Motions has been used. We see that if we are given the values of time(t), acceleration(a), initial velocity(u) , finaly velocity (v) and displacement(d), the we can apply the values and satisfy the equation to get our solution.

Let's do it !!

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★ Formula Used :-

$\: \\ \large{\boxed{\boxed{\sf{v^{2} \: \: - \: \: u^{2} \: \: = \: \: \bf{2as}}}}}$

This is the Third Equation of Motion.

$\: \\ \large{\boxed{\boxed{\sf{s \: = \: \bf{ut \: \: + \: \: \dfrac{1}{2} \: \times \: at^{2}}}}}}$

This is the Second Equation of Motion.

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Solution :-

1.) A body moving with an initial velocity of 1 m/s accelerates uniformly at 0.5 m/s2. Find out its velocity when it has a displacement of 48 m?

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Answer.) Given,

» Initial Velocity of the body = u = 1 m/sec

» Acceleration of the body = a = 0.5 m/sec²

» Displacement done by body = s = 48 m

• Let the final velocity of the body be 'v' m/sec.

Then, by using the Third Equation of Motion, we get,

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: v^{2} \: \: - \: \: u^{2} \: \: = \: \: \bf{2as}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: v^{2} \: \: - \: \: (1 \: m\:sec^{-1})^{2} \: \: = \: \: \bf{2 \: \times \: 0.5 \: m\:sec^{-2} \: \times \: 48 \: m}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: v^{2} \: \: - \: \: (1 \: m^{2}\:sec^{-2}) \: \: = \: \: \bf{48 \: m^{2}\:sec^{-2}}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: v^{2} \: \: \: \: = \: \: \bf{48 \: m^{2}\:sec^{-2}} \: \: + \: \: (1 \: m^{2}\:sec^{-2})\: \: = \: \: 49 \: \: m^{2}\:sec^{-2}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: v \: \: \: \: = \: \: \bf{\sqrt{49 \: \: m^{2}\:sec^{-2}} \: \: = \: \: \underline{\underline{7 \: \: m\:sec^{-1}}}}}}$

$\: \\ \large{\underline{\underline{\rm{\mapsto \: \: \: Thus, \: the \: final \: velocity \: of \: the \: body \: is \: \: \boxed{\bf{7 \;\: m\:sec^{-1}}}}}}}$

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2.) For a body moving with an initial velocity u and uniform acceleration a. Find the displacement of the body in time t.

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Answer.) Given,

» Initial velocity of the body = u

» Acceleration of the body = a

» Time taken by the body = t

Here, we can use the Position - Time Relationship to find our answer.

By Second Equation of Motion we know that,

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: s \: = \: \bf{ut \: \: + \: \: \dfrac{1}{2} \: \times \: at^{2}}}}$

Hence this is the required answer.

$\: \\ \large{\underline{\underline{\rm{\mapsto \: \: \: Thus, \: the \: displacement \: \bf{\bf{s}} \: \rm{of \: the \: body \: in \: time \: t \: is} \: \: \boxed{\bf{ut \: + \: \dfrac{1}{2}at^{2}}}}}}}$

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$\: \\ \: \qquad \large{\underbrace{\underbrace{\sf{More \: \: to \: \: know \: \: :-}}}}$

$\: \\ \leadsto \: \: \sf{v \: - \: u \: \: = \: \: at}$

This is the First Equation of Motion.

It is also know as Velocity Position Relationship.

$\: \\ \leadsto \: \: \sf{Velocity \: = \: \dfrac{Displacement}{Time}}$

$\: \\ \leadsto \: \: \sf{Acceleation \: = \: \dfrac{v \: - \: u}{t}}$

$\: \\ \leadsto \: \: \sf{Displacement \: = \: Velocity \: \times \: Time}$

$\: \\ \leadsto \: \: \sf{s_{n_{(th)}} \: = \: u + \: \dfrac{a}{2}(2n - 1)}$

Answer 2

Solution for 1 :

Given,

A body moving with an initial velocity of 1 m/s accelerates uniformly at 0.5 m/s2.

To find :

Find out its velocity when it has a displacement of 48 m.

Solution :

Given, u = 1 m/s ; a = 0.5 m/s² ; s = 48 m

Now using 3rd equation of motion :

⇒ v² = u² + 2as

⇒ v² = 1² + 2 * 0.5 * 48

⇒ v² = 1 + 48

⇒ v² = 49

⇒ v = √49

⇒ v = 7 m/s

∴ Velocity of body at displacement of 48 m = 7 m/s

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Solution for 2 :

Given,

Initial velocity = u m/s

Acceleration = a m/s²

Time = t s

To find :

Displacement of body

Solution :

At first, considering that there is no acceleration, so :

⇒ Displacement , s = Velocity * time

⇒ s = V * t                     _(i)

Now we know, average velocity is the mean of initial and final velocity.

⇒ V = (u + v)/t

Now, from first equation of motion, v = u + at

⇒ V = (u + u + at)/2

⇒ V = (2u + at)/2

⇒ V = 2u/2 + at/2

⇒ V = u + 1/2 at               _(ii)

Now putting value of (ii) in (i) :

⇒ s = (u + 1/2 at) * t

⇒ s = ut + 1/2 at²

∴ Displacement of body in time t , s= ut + 1/2 at²