Chapter 1, Problem 1PP

(Conversion) Determine which of the following equations can’t be valid because they yield incorrect unit measurements:

$\begin{array}{l}\text{a}\text{. F = m a}\hfill \\ \text{b}\text{. F = m}\left({\text{v}}^{\text{2}}\text{/t}\right)\hfill \\ \text{c}\text{. d = ½ }\left({\text{a t}}^{\text{2}}\right)\hfill \\ \text{d}\text{. d = v t}\hfill \\ \text{e}\text{. F = mvt}\hfill \\ \text{F = force }\left(\text{N}\right)\hfill \\ \text{m = mass }\left(\text{kg}\right)\hfill \\ \text{a = acceleration }\left({\text{m/s}}^{\text{2}}\right)\hfill \\ \text{v = velocity }\left(\text{m/s}\right)\hfill \\ \text{t = time }\left(\text{s}\right)\hfill \end{array}$

Program Plan Intro

To determine the unit of measurements of the given equation, F=ma is valid or not.

Program Description Answer

F= ma is a valid equation.

Explanation of Solution

Given information:

$F=ma$

Here,

F= Force(N)

m= mass (kg)

a = acceleration (m/s²)

v = velocity (m/s)

t = time (s)

Explanation:

Solving the given equation,

$\begin{array}{l}F=ma\hfill \\ N=kg\left(m/s^2\right)\hfill \end{array}$

It is a valid equation.

As, 1 Newton= $kgm{s}^{-2}$

Program Plan Intro

To determine the unit of measurements of the given equation, F = m(v²/t) is valid or not.

Program Description Answer

$F=m\left(\frac{v^2}{t}\right)$ is not a valid equation.

Explanation of Solution

Given information:

$F=m\left(\frac{v^2}{t}\right)$

Here,

F= Force(N)

m= mass (kg)

a = acceleration (m/s²)

v = velocity (m/s)

t = time (s)

Explanation:

Solving the given equation,

$\begin{array}{l}F=m\left(\frac{v^2}{t}\right)\\ N=kg\left(\frac{{\left(m/s\right)}^2}{s}\right)\\ N=kg\left(\frac{m^2}{s^3}\right)\end{array}$

It is a not a valid equation.

As, 1 Newton= $kgm{s}^{-2}$ .

Program Plan Intro

To determine the unit of measurements of the given equation, $\text{d }=\frac{1}{2}\left(a{t}^2\right)$ is valid or not.

Program Description Answer

$\text{d }=\frac{1}{2}\left(a{t}^2\right)$ is a valid equation.

Explanation of Solution

Given information:

$\text{d }=\frac{1}{2}\left(a{t}^2\right)$

Here,

a = acceleration (m/s²)

t = time (s)

d = distance (meter)

Explanation:

Solving the given equation,

$\begin{array}{l}\text{d }=\frac{1}{2}\left(a{t}^2\right)\\ m=\frac{1}{2}\left(\frac{m{s}^2}{s^2}\right)\\ m=\frac{1}{2}\left(\frac{m\cancel{s^2}}{\cancel{s^2}}\right)\\ m=\frac{1}{2}m\end{array}$

It is a valid equation.

As, distance is measured in meter(m).

Program Plan Intro

To determine the unit of measurements of the given equation, $\text{d }=\text{ vt}$ is valid or not.

Program Description Answer

$\text{d }=\text{ vt}$ is a valid equation.

Explanation of Solution

Given information:

$\text{d }=\text{ vt}$

Here,

v = velocity (m/s)

t = time (s)

d = distance (meter)

Explanation:

Distance is measured in meter(m).

Solving the given equation,

$\begin{array}{l}\text{d }=\text{ vt}\\ m=\frac{ms}{s}\\ m=\frac{m\cancel{s}}{\cancel{s}}\\ m=m\end{array}$

It is a valid equation.

As, distance is measured in meter(m).

Program Plan Intro

To determine the unit of measurements of the given equation, $\text{F }=\text{ mvt}$ is valid or not.

Program Description Answer

$\text{F }=\text{ mvt}$ is not a valid equation.

Explanation of Solution

Given information:

$\text{F }=\text{ mvt}$

Here,

F= Force(N)

m= mass (kg)

v = velocity (m/s)

t = time (s)

Explanation:

Solving the given equation,

$\text{F }=\text{ mvt}$

Force is measured in newton(N).

$\begin{array}{l}F=\text{ mvt}\\ N=\frac{kg\text{ ms}}{s}\\ N=\frac{kg\text{ m}\cancel{s}}{\cancel{s}}\\ N=kgm\end{array}$

It is a not a valid equation.

As, 1 Newton= $kgm{s}^{-2}$ .