How Does Mass Affect Momentum and Kinetic Energy in Motion?

[senseandsanity]

I need help with this question:
A cardinal of mass 3.60×10-2 kg and a baseball of mass 0.141 kg have the same kinetic energy. What is the ratio of the cardinal's magnitude of momentum to the magnitude of the baseball's momentum (p_c/p_b)?

 

[Sirus]

Consider that
|\vec{p}|=m|\vec{v}| and
E_{K}=\frac{1}{2}mv^2
Can you figure it out from there?

 

[wais]

In order to determine the ratio of the cardinal's momentum to the baseball's momentum, we first need to understand the relationship between momentum and kinetic energy.

Momentum is defined as the product of an object's mass and velocity, and it is a vector quantity with both magnitude and direction. On the other hand, kinetic energy is the energy an object possesses due to its motion, and it is a scalar quantity with only magnitude.

In this scenario, both the cardinal and the baseball have the same kinetic energy, meaning they have the same amount of energy due to their motion. However, the cardinal has a much smaller mass compared to the baseball. This means that the cardinal must have a higher velocity in order to have the same kinetic energy as the baseball.

Since momentum is directly proportional to mass and velocity, we can conclude that the cardinal must have a higher momentum than the baseball due to its higher velocity. This can be expressed as p_c > p_b.

To find the exact ratio of the cardinal's momentum to the baseball's momentum, we can use the equation for momentum (p = mv) and substitute in the given values:

p_c = m_c * v_c
p_b = m_b * v_b

Since we know that the kinetic energy (KE) is the same for both objects, we can set their kinetic energy equations equal to each other:

KE_c = 1/2 * m_c * v_c^2
KE_b = 1/2 * m_b * v_b^2

Since the kinetic energy is the same, we can set these equations equal to each other and solve for the ratio of velocities:

1/2 * m_c * v_c^2 = 1/2 * m_b * v_b^2
v_c^2 = (m_b/m_c) * v_b^2
v_c/v_b = √(m_b/m_c)

Now, we can substitute this ratio of velocities into the momentum equation to find the ratio of the cardinal's momentum to the baseball's momentum:

(p_c/p_b) = (m_c * v_c)/(m_b * v_b)
(p_c/p_b) = (m_c/m_b) * (v_c/v_b)
(p_c/p_b) = (m_c/m_b) * √(m_b/m_c)

Finally, we can plug in the given masses for the cardinal and the baseball to find the numerical value of the ratio:

(p_c/p