# Question 6028a

Jun 17, 2015

The steel ball will have a higher momentum.

#### Explanation:

Momentum depends on the velocity and on the mass of an object according to the following relationship

$p = m \cdot v$, where

$p$ - momentum;
$m$ - the mass of the object;
$v$ the velocity of the object.

In your case, you know that the speed of the steel ball is equal to the speed of the cotton ball, so you can write

${p}_{\text{steel" = m_"steel}} \cdot v$

and

${p}_{\text{cotton" = m_"cotton}} \cdot v$

You can establish a relationship between the momentum of the steel ball and the momentum of the cotton ball by dividing these two equations

${p}_{\text{steel"/p_"cotton" = m_"steel"/m_"cotton}} \cdot \frac{\cancel{v}}{\cancel{v}}$

However, you have no information about the masses of the two balls, but you know something about their volumes.

Mass can be expressed by using density and volume

$\rho = \frac{m}{V} \implies m = \rho \cdot V$

Since you know that the two balls have equal volumes, you can write

p_"steel"/p_"cotton" = (rho_"steel" * cancel(V))/(rho_"cotton" * cancel(V))#

${p}_{\text{steel"/p_"cotton" = rho_"steel"/rho_"cotton}}$

Since the density of steel is bigger than the density of cotton, you have

${p}_{\text{steel" = underbrace(rho_"steel"/rho_"cotton")_(color(blue)(">0")) * p_"cotton}}$

Therefore, the steel ball will have the greater momentum if velocity and volume are equal.