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# A billiard ball moving with a speed of $5m {{s} ^ {-1}}$ collides with an identical ball originally at rest. If the first ball stops after collision then the sound ball will move forward with a speed of A) $10m{{s}^{-1}}$B) $5m{{s}^{-1}}$C) $2.5m {{s}^{-1}}$ D) $1.0m {{s}^{-1}}$

Last updated date: 22nd Jul 2024
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Hint: Momentum is defined as the product of the mass and velocity of objects. Momentum is a vector quantity which has both magnitude and direction. Momentum (p) =mv and the dimensional formula of momentum is $ML{{T}^{-1}}$. The SI unit of momentum is kilogram meter per second $(kgm{{s}^{-1}})$ and momentum is a conserved quantity.

Complete step-by-step solution:
Speed is defined as the distance covered per unit time and speed is a scalar quantity. SI unit of speed is $m{{s}^{-1}}$
Speed is given as:
$v=\dfrac{d}{t}$
Where d = distance
t= time
$v$ =speed
Velocity is defined as the rate of change of displacement with respect to time and in kinematics velocity is a fundamental concept.SI unit of velocity is $m{{s}^{-1}}$ velocity tracking is the measure of velocity. Velocity is also a vector quantity.
Velocity (v) =$\dfrac{\Delta S}{\Delta t}$
The dimensional formula of velocity is ${{M}^{0}}{{L}^{1}}{{T}^{-1}}$
From the data we can observe that there is no external force. So we can apply momentum conservation,
${{p}_{1}}=mV=5m$
\begin{align} & {{p}_{2}}=0 \\ & {{p}_{1}}=5m+0=5m \\ \end{align}
Law of conservation of momentum states that for two or more bodies in an isolated system acting upon each other, unless an external force is applied the total momentum of their bodies remains constant.
\begin{align} & {{p}_{f}}=m{{\nu }_{1}}+m{{\nu }_{2}} \\ & {{p}_{f}}=m\times 0+m{{\nu }_{2}} \\ \end{align}
\begin{align} & {{p}_{i}}={{p}_{f}} \\ & 5m=m{{\nu }_{2}} \\ & {{\nu }_{2}}=5m{{s}^{-1}} \\ \end{align}
So the correct option is B
Due to elastic collision of bodies having equal mass, their velocities get interchanged.

Note: Students, the solution to this problem is also obtained by utilizing the equation of motion and also by the speed and direction relationship. The direction can be specified or mentioned in plus or minus sign in one dimensional motion and in order to know the direction of motion Newton’ s law of motion is very helpful.