Question

A particle moving with a uniform acceleration travels $24m$ and $64m$in the first two constant intervals of $4\sec$. Its initial velocity is
A. $1m/\sec$
B. $10m/\sec$
C. $5m/\sec$
D. $2.5m/\sec$

Answer 1

The equation of motions describes the basic concept of the object’s motion like position, acceleration, and velocity. These equations govern the object’s motion in $1D,2D{\text{ and }}3D$. To solve the given problem, consider the equation of the motions.

Formula used:
Formula to find the initial velocity:
$\Rightarrow v = u + at$
Where, $v$ is the velocity, $a$is the acceleration, $t$ is the time, and $u$ is the initial velocity.

Complete step by step answer:
A particle is moving with uniform acceleration. The particle has two constant intervals of $4\sec$at
$24m$ and at $64m$.
We need to calculate the velocity of each interval first. Initial velocity for the particle at $24m$ of constant interval of $4\sec$.
$\Rightarrow v = u + at$
Where, $v$ is the velocity, $a$ is the acceleration, $t$ is the time, and $u$ is the initial velocity.
Substitute the values $t = 4\sec$ in the formula.
$\Rightarrow v = u + 4a$
The value for the velocity is the same for the next interval at $64m$. That is, $v = u + 4a$
Next, calculate the speed. The values for the speed are given.
For first $4\sec$the speed is,
$\Rightarrow s = ut + \dfrac{1}{2}a{t^2}$
Substitute the Values,
$\Rightarrow 24 = 4u + \dfrac{1}{2}a \times {4^2}$
$\Rightarrow 24 = 4u + \dfrac{1}{2}16a$
Simplify the given equation.
$\Rightarrow 24 = 4u + 8a$
$\Rightarrow 4 + 2a = 6$---- 1
For a second $4\sec$the speed is,
$\Rightarrow s = ut + \dfrac{1}{2}a{t^2}$
Substitute the Values,
$\Rightarrow 64 = \left( {u + 4a} \right)4 + \dfrac{1}{2}a{4^2}$
Simplify the equation,
$\Rightarrow 16 = \left( {u + 4a} \right) + 2a$
$\Rightarrow 4 + 6a = 16$--- 2
Subtract equation 2 from 1
We get, $4a = 10$therefore, $a = 2.5$substitute the value of acceleration in equation 1.
$\Rightarrow u + 2a = 6$
$\Rightarrow u + 2\left( {2.5} \right) = 6$
$\Rightarrow u = \dfrac{6}{6}$
$\Rightarrow u = 1$
Therefore, the initial value is $1m/\sec$.

So, the correct answer is “Option A”.

Note:
The equation of the motion used to solve the problem is called the first equation of motion. There are totally three types of motion. Equations of the motions are used to solve the kinematics problems. We have to identify the parameters given and then we have to choose the appropriate equations. It is also used to solve the calculations of the optical properties.