Question

Which of the following is a scalar quantity? Mass, velocity, Force, acceleration.

Answer 1

Hint: There are two types of physical quantity. Scalar quantity and vector quantity. Scalar quantity is physical quantity which has magnitude but they don’t have direction. On the other hand vector quantities are the physical quantities which have both magnitude and direction. Among four given physical quantities see which of them have both magnitude and direction and which of them has only magnitude.

Formula used:
$\begin{array}{rl}& \text{Velocity(}v\text{)=}{\displaystyle \frac{\text{Change in displacement}}{\text{Change in time}}}={\displaystyle \frac{ds}{dt}}\\ & \text{Acceleration(}a\text{)=}{\displaystyle \frac{\text{Change in velocity}}{\text{Change in time}}}={\displaystyle \frac{dv}{dt}}={\displaystyle \frac{d^{2}s}{d{t}^2}}\\ & \text{Force=}ma=m{\displaystyle \frac{d^{2}s}{d{t}^2}}\\ & {\displaystyle \frac{d}{dx}}\left(UV\right)=U{\displaystyle \frac{dV}{dx}}+V{\displaystyle \frac{dU}{dx}}\end{array}$

Complete step-by-step answer:
Velocity of a particle is defined as the rate of change of displacement. If the change in displacement $ds$ occurs over a time $dt$, then velocity $v={\displaystyle \frac{ds}{dt}}$
As displacement is a vector quantity i.e displacement of a particle obeys vector laws of addition and the time is not a vector quantity because it is added algebraically. So velocity will be a vector quantity.
Acceleration of a particle is defined as the rate of change of velocity. So $a={\displaystyle \frac{dv}{dt}}={\displaystyle \frac{d^{2}s}{d{t}^2}}$. As velocity is a vector quantity acceleration would also be a vector quantity.
Mass of an object is defined by the amount of matter it contains. So when two masses added it will add up algebraically so the mass of an object has only magnitude not direction. So mass is a scalar quantity.
The quantitative definition of force is given by Newton’s 2nd law, which states that Force acting upon an object is directly proportional to the rate of change of its momentum.
If momentum of a object is $p$then force is given by,$\begin{array}{rl}& F={\displaystyle \frac{dp}{dt}}={\displaystyle \frac{d(mv)}{dt}}=m{\displaystyle \frac{dv}{dt}}+v{\displaystyle \frac{dm}{dt}}=m{\displaystyle \frac{dv}{dt}}(\because {\displaystyle \frac{dm}{dt}}=0)\\ & \Rightarrow F=ma\end{array}$
As acceleration is a vector quantity and mass is not a vector quantity so force will be a vector quantity.
So only mass is a scalar quantity among the given 4 quantities.

Note: Every vector is represented by an arrowhead over the magnitude. i.e. if $A$ is the magnitude of a vector then it is represented by $\overrightarrow{A}$. Every vector obey the laws of vector addition and subtraction. If a quantity doesn’t obey vector addition is called scalar. So scalars are added algebraically.