Question

Are the units of the formula ma = mv2/2 dimensionally consistent? Select the single best answer.

(A) yes – both ma and 1/2mv2 have units of kg•m2/s2
(B) no – ma has units of kg•m/s2 and 1/2mv2 has units of kg•m2/s2
(C) no – ma has units of kg•m2/s2 and 1/2mv2 has units of kg•m/s2
(D) no – ma has units of kg•m/s and 1/2mv2 has units of kg•m2/s2
(E) yes – both ma and 1/2mv2 have units of kg•m/s2
(F) yes – the units of m cancel on both sides.
(G) yes – both ma and 1/2mv2 have units of kg•m/s

Answer 1

To solve the problem we will simply perform equivalence between both expressions. We will proceed to place your units and develop your internal operations in case there is any. From there we will compare and look at its consistency

$ma = \text{Mass}\times \text{Acceleration}$

$ma = kg \cdot \frac{m}{s^2}$

At the same time we have that

$\frac{1}{2}mv^2 = \text{Mass}\times \text{Velocity}^2$

$\frac{1}{2}mv^2 = kg ( \frac{m}{s})^2$

$\frac{1}{2}mv^2 = kg \cdot \frac{m^2}{s^2}$

Therefore there is not have same units and both are not consistent and the correct answer is B.

Answer 2

The units of the given formula are not consistent because ma has units of kg•m/s2 and 1/2mv2 has units of kg•m2/s2. Hence, option (B) is correct.

Dimensional consistency is the method to check the equality of the units of an equation. A mathematical expression for any quantity is said to be dimensionally consistent only when it has the same dimensional unit at both sides.

As per the given problem, the expression is given as,

$ma = \dfrac{1}{2}mv^{2}$

Here,

The term on the right-hand side of the given expression is ma. Which is nothing but Force. And the dimensional unit of force is Newtons (N).

Force (F) = ma

F = kg - m/s²

While the term on the left-hand side of the given expression is known as kinetic energy, whose dimensional unit is Joules (J).

$KE =\dfrac{1}{2}mv^{2}\\\\ KE =\;\rm kg -m^{2}/s^{2}$

So, each of the given terms is having different units for which the overall expression is not consistent.

Thus, we can conclude that the units of the given formula are not consistent because ma has units of kg•m/s2 and 1/2mv2 has units of kg•m2/s2. Hence, option (B) is correct.

Learn more about the Dimensional consistency here:

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