Question

Consider two objects (Object 1 and Object 2) moving in the same direction on a frictionless surface. Object 1 moves with speed v1 = v and has mass m1 = 2m. Object 2 moves with speed v2 = √2v and has mass m2 = m.Part A:Which object has the larger momentum?A) Object 1 has the greater momentum.B) Object 2 has the greater momentum.C) Both objects have the same momentum.Part B:Which object has the larger kinetic energy?Object 1 has the greater kinetic energy.Object 2 has the greater kinetic energy.The objects have the same kinetic energy.

Answer 1

1) A) Object 1 has the greater momentum

The magnitude of the momentum of an object is given by:

$p=mv$

where

m is the mass of the object

v is its speed

Object 1 has a mass of $m_1 = 2m$ and a speed of $v_1 = v$, so its momentum is

$p_1 = m_1 v_1 = (2m)(v)=2mv$

Object 2 has a mass of $m_2 = m$ and a speed of $v_2 = \sqrt{2} v$, so its momentum is

$p_2 = m_2 v_2 = (m)(\sqrt{2} v)=\sqrt{2}mv$

So we see that $p_1 > p_2$, so object 1 has the greater momentum.

2) The objects have the same kinetic energy.

The kinetic energy of an object is given by

$K=\frac{1}{2}mv^2$

where

m is the mass of the object

v is its speed

Object 1 has a mass of $m_1 = 2m$ and a speed of $v_1 = v$, so its kinetic energy is

$K_1 = \frac{1}{2}m_1 v_1^2 = \frac{1}{2}(2m)(v)^2=mv^2$

Object 2 has a mass of $m_2 = m$ and a speed of $v_2 = \sqrt{2} v$, so its kinetic energy is

$K_2 = \frac{1}{2}m_2 v_2^2 = \frac{1}{2}(m)(\sqrt{2} v)^2=mv^2$

So we see that $K_1 =K_2$, so the objects have same kinetic energy