Question

Moving with uniform acceleration, a body covers $150\,m$ during $10$ seconds so that it covers $24\,m$ during tenth second. Find initial velocity and acceleration of the body.
A. \[2\,m{s^{ - 1}};5\,m{s^{ - 2}}\]
B. \[5\,m{s^{ - 1}};2\,m{s^{ - 2}}\]
C. \[3\,m{s^{ - 1}};4\,m{s^{ - 2}}\]
D. \[4\,m{s^{ - 1}};3\,m{s^{ - 2}}\]

Answer 1

Hint:To solve this question we should have a basic idea that it is about the motion of a body in a straight line. Hence we will solve this question using equations of motion. First we will find acceleration and then velocity. We will use the 2nd equation of motion to find a relation between acceleration and initial velocity and then using formula for distance covered in nth second we will find acceleration.

Formula used:
\[v = u + at\]
where \[u\]=initial velocity, \[a\]=acceleration and \[v\]=final velocity
\[s = ut + \dfrac{1}{2}a{t^2}\]
Where \[s\]=distance travelled by the body, \[t\]=time and \[u\]=initial velocity.
\[{s_n} = u + \dfrac{a}{2}\left( {2n - 1} \right)\]
Where \[{s_n}\] is distance travelled by a body in nth second, \[a\]=acceleration, \[u\]=initial velocity, n=nth second.

Complete step by step answer:
Let the body have initial velocity $u$, acceleration $a$, final velocity $v$ and may it cover $s$ distance and sn distance in time $t$. Let us start solving this question by first finding relation between the acceleration and initial velocity of the body using the 2nd equation of motion:
\[s = ut + \dfrac{1}{2}a{t^2}\]
let us substitute the values from the question
\[150 = u10 + \dfrac{1}{2}a{10^2}\]
\[\Rightarrow 150 = 10u + \dfrac{1}{2} \times a \times 100\]
\[\Rightarrow 150 = 10u + 50a\]
Divide the whole equation by 5.
\[30 = 2u + 10a\]
\[\Rightarrow 30 - 10a = 2u..........(1)\]

Now let us find acceleration using the formula for distance covered in 10th second,
\[{s_n} = u + \dfrac{a}{2}\left( {2n - 1} \right)\]let us substitute the values;
\[\Rightarrow 24 = u + \dfrac{a}{2}\left( {2 \times 10 - 1} \right)\]
\[\Rightarrow 24 = u + \dfrac{a}{2} \times \left( {19} \right)\]
Multiply both sides by 2:
\[48 = 2u + a\left( {19} \right)\]
\[\Rightarrow 48 = 2u + 19a\]................(2)
Now lets substitute the value of 2u from equation 1 into 2
\[48 = 30 - 10a + 19a\]
\[\Rightarrow 48 - 30 = 19a - 10a\]
\[\Rightarrow 18 = 9a\]
\[\therefore a = 2\,m{s^{ - 2}}\]

Now let's find initial velocity using equation 1
\[30 - 10a = u2\]
\[\Rightarrow 30 - 10\left( 2 \right) = u2\]
\[\Rightarrow 30 - 20 = u2\]
\[\Rightarrow 2u = 10\]
\[\Rightarrow u = \dfrac{{10}}{2}\,m{s^{ - 1}}\]
\[\therefore u = 5\,m{s^{ - 1}}\]

Hence the correct answer is option B.

Note:Newton's equations of motion can be applied only when the body is moving with uniform acceleration. In the above questions the body is moving with uniform acceleration so we have applied Newton's equations. In some cases, when the acceleration is variable then we need to apply calculus for solving those problems.