Question

A system consists of two moving objects, each represented by a dot and an arrow in the diagram. What is the magnitude and direction of the net momentum of the system? Express the direction of the net momentum as a counterclockwise angle to the +x-axis.

Answer 1

let p_x be the momentum in x-direction & p_y be the momentum in y-direction respectively.

assuming that,

mass of the first object on x-direction, m_x = 5 kg

mass of the second object on y-direction, m_y = 8 kg

velocity of the object on x-axis, v_x = 25 m/s

velocity of the object on y-axis, v_y = 15 m/s

the net momentum of the system is given as ::

P_net = p_x + p_y

or P_net = m_x v_x + m_y v_y                             { eq, 1 }

inserting the values in above eq.

P_net = (5 kg) (25 m/s) + (8 kg) (15 m/s)

P_net = (200 kg.m/s) + (120 kg.m/s)

magnitude of the net momentum of the system is given as ::

P_net = p_x² + p_y²                                  { eq. 2 }

inserting the value of ' p_x' & ' p_y' in eq.2,

P_net = (200 kg.m/s)² + (120 kg.m/s)²

P_net = 54400 (kg.m/s)²

P_net = 233.2 kg.m/s

direction of the net momentum of the system is given as ::

= tan^-1 (p_y / p_x)    { eq. 3 }

inserting the values in eq.3,

= tan^-1 [(120 kg.m/s) / (200 kg.m/s)]

= tan^-1 (0.6)

= 30.9 degree

expression for the direction of the net momentum as a counterclockwise angle to the x-axis is given as ::

p_x = P_net Cos                { eq. 4 }

where, = 150 degree

inserting the values in eq.4,

p_x = (233.2 kg.m/s) Cos (150 degree)

p_x = - 201.9 kg.m/s