Question

An object has a kinetic energy of 284 J and a momentum of magnitude 23.5 kg · m/s.

(a) Find the speed of the object.
m/s

(b) Find the mass of the object.
kg

Answer 1

Concepts and reason

The concept used to solve this problem is related energy and momentum of an object.

First, eliminate the mass by dividing the expression of kinetic energy by momentum. It gives the velocity of the object. Finally, rearrange the expression of momentum for mass and calculate it.

Fundamentals

The expression of kinetic energy of a moving object is given by,

$K = \frac{1}{2}m{v^2}$

Here, $m$is mass of the object,$v$ is the velocity.

The expression of kinetic energy of a moving object is given by,

$p = mv$

Here, $m$is mass of the object,$v$ is the velocity.

(a)

Calculate the velocity of the object.

The expression of kinetic energy of a moving object is given by,

$K = \frac{1}{2}m{v^2}$

Here, $m$is mass of the object,$v$ is the velocity.

The expression of kinetic energy of a moving object is given by,

$p = mv$

Here, $m$is mass of the object,$v$ is the velocity.

Divide the expression $K = \frac{1}{2}m{v^2}$ by$p = mv$,

$\begin{array}{c}\\\frac{K}{p} = \frac{{\frac{1}{2}m{v^2}}}{{mv}}\\\\ = \frac{1}{2}v\\\end{array}$

Rearrange it for$v$,

$v = \frac{{2K}}{p}$

Substitute $284{\rm{ J}}$ for $K$ and $23.5{\rm{ kg}} \cdot {\rm{m}} \cdot {{\rm{s}}^{ - 1}}$ for$p$,

$\begin{array}{c}\\v = \frac{{2\left( {284{\rm{ J}}} \right)}}{{23.5{\rm{ kg}} \cdot {\rm{m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 24.2{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}\\\end{array}$

(b)

Calculate the mass of the object.

Rearrange the expression of momentum as follows,

$m = \frac{p}{v}$

Substitute $23.5{\rm{ kg}} \cdot {\rm{m}} \cdot {{\rm{s}}^{ - 1}}$ for $p$ and $24.2{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}$ for$v$,

$\begin{array}{c}\\m = \frac{{23.5{\rm{ kg}} \cdot {\rm{m}} \cdot {{\rm{s}}^{ - 1}}}}{{24.2{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 0.971{\rm{ kg}}\\\end{array}$

Ans: Part a

The velocity of the object is$24.2{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}$.