Question

The distance travelled by a body in the nth second is given by the expression$\left( {2 + 3n} \right)$. Find the initial velocity and acceleration. Also, find its velocity at the end of $2$ seconds.

Answer 1

Separate the known and unknown terms and compare the equations to find out the unknown required terms.

Complete step by step answer:
Let us consider the distance travelled by an object in nth second $= n + \dfrac{1}{2}a(2n - 1)$
Replace, $n = u$ in the above equation and equate $(2 + 3n)$
Therefore,
$2 + 3n = u + \dfrac{1}{2}a(2n - 1)$
Now, open the brackets and compare both the sides of the equation

$$2 + 3n = u + \dfrac{1}{2}a(2n) - (1)\dfrac{a}{2} \\\ 2 + 3n = u + an - \dfrac{a}{2} \\\$$

Implies –
$3n = an$ and $2 = u - \dfrac{a}{2}$
$3n = an$
Same terms in the multiplicative form on both sides of the equation cancel each other.
$a = 3m/{s^2}$
Now, take $2 = u - \dfrac{a}{2}$
Place the value of “a” in the above equation
$2 = u - \dfrac{3}{2}$
Simplify the above equation –
When terms are moved from its side, the sign of the terms also changes. Positive changes to negative and negative changes to positive.
$2 + \dfrac{3}{2} = u$
Make unknown “U” as the subject –
$u = 2 + \dfrac{3}{2}$
Take LCM
$u = \dfrac{{4 + 3}}{2} \\\ u = \dfrac{7}{2} \\\ u = 3.5m/s \\\$
Now, velocity after $2$ Seconds is $V = u + at$
Place the values of initial velocity (u) and the acceleration (a)
$V = u + at \\\ V = 3.5 + 3(2) \\\ V = 3.5 + 6 \\\ V = 9.5m/s \\\$
Thus, the required solution is –
Initial Acceleration, $a = 3{\text{ }}m/{s^2}$
Initial Velocity, $u = 3.5\,{\text{ m/}}{{\text{s}}^{}}$
Velocity after $2$ seconds, $V = 9.5\;{\text{m/s}}$

Note:
The velocity of an object is the rate of change of its position with respect to time. In Simple terms, velocity is also known as the speed and is distance travelled upon the time taken. It can be expressed as meter per second or kilometre per hour. Acceleration is the vector quantity and is the rate of change of the velocity of an object with respect to time. Acceleration can be expressed as a meter per Second Square.