A ball with a mass of # 5 kg# is rolling at #5 m/s# and elastically collides with a resting ball with a mass of #2 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of # 5    kg# is rolling at #5   m/s# and elastically collides with a resting ball with a mass of #2  kg#. What are the post-collision velocities of the balls?

Charlotte Aaron · Accepted Answer

#"The answer " vec v_("2_after")=50/7" "m/s" , "vec v_("1_after")=15/7" "m/s# #"The definition of the momentum is given as " vec P=m*vec v# #"Where ;"# #"m is mass of object and v is its velocity"# #"if object has not a velocity,it will not have any momentum"# #(vec P=m*0=0)# #"The momentum of any object is a vectorial quantity,"# #"it has a magnitude and direction."# #"The momentum problems is solved by conservation of momentum"# #"let us find momentums of object for before impact"# #"........................................................................................."# #vec P_("1_before")=m_1* vec v_("1_before")# #"(momentum before impact for a mass of 5 kg)"# #"plug m=5 kg and v=5 "m/s# #vec P_("1_before")=5*5=25 " "kg*m/s# #vec P_("2_before")=m_2* vec v_("2_before")# #"(momentum before impact for a mass of 2 kg)"# #"plug m=2 kg and v=0 "m/s# #vec P_("2_before")=2*0=0 " "kg*m/s# #"The vectorial sum of the momentums before impact is"# # color(red)(Sigma vec P_("before"))=vec P_("1_before")+vec P_("2_before")# #"if collision is centered, the total momentum"# #"will be "color(red)(Sigma vec P_("before"))=25+0=25 " "kg*m/s# #"now let us find the momentums of objects for after impact"# #.........................................................................................# #vec P_("1_after")=m_1* vec v_("1_after")# #"(momentum after impact for a mass of 5 kg)"# #vec P_("1_after")=5*vec v_("1_after") # #vec P_("2_after")=m_2* vec v_("2_after")# #"(momentum after impact for a mass of 2 kg)"# #vec P_("2_after")=2*vec v_("2_after") # #"The vectorial sum of the momentums after impact is"# #color(blue)(Sigma vec P_("after"))=vec P_("1_after")+vec P_("2_after")# #color(blue)(Sigma vec P_("after"))=5*vec v_("1_after") +2*vec v_("2_after") # #"Let us write the conservation of momentum"# #.............................................................................# #color(red)(Sigma vec P_("before"))=color(blue)(Sigma vec P_("after"))# #25=5*vec v_("1_after") +2*vec v_("2_after")" " "(1)" # #vec v_("1_before")+ vec v_("1_after")=vec v_("2_before")+vec v_("2_after")" "(2)# #"note: you can proof the equation (2) using together momentum "##"and kinetic energy conservations equations. "# #5+vec v_("1_after")=0+vec v_("2_after")" "(3)# #vec v_("2_after")=5+vec v_("1_after")# #"now let us use (1)"# #25=5*vec v_("1_after") +2(5+vec v_("1_after"))# #25=5*vec v_("1_after") +10+2*vec v_("1_after")# #25-10=7*vec v_("1_after")# #15=7*vec v_("1_after")# #vec v_("1_after")=15/7" "m/s# #"use (3)"# #5+15/7=vec v_("2_after")# #vec v_("2_after")=50/7" "m/s#

Layla Ahmed · Answer

undefined To find the post-collision velocities of the balls, we can use the principle of conservation of momentum and kinetic energy. According to this principle, the total momentum and kinetic energy before the collision should be equal to the total momentum and kinetic energy after the collision.

Given:
Mass of ball 1 (m1) = 5 kg
Initial velocity of ball 1 (u1) = 5 m/s
Mass of ball 2 (m2) = 2 kg
Initial velocity of ball 2 (u2) = 0 m/s (since the ball is at rest)

Using conservation of momentum:
m1 * u1 + m2 * u2 = m1 * v1 + m2 * v2

Using conservation of kinetic energy (since the collision is elastic):
0.5 * m1 * u1^2 + 0.5 * m2 * u2^2 = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2

We have two equations and two unknowns (v1 and v2), which we can solve simultaneously to find the post-collision velocities of the balls. Plugging in the given values and solving the equations will give us the post-collision velocities.

After solving the equations, we find:
v1 ≈ 1.364 m/s
v2 ≈ 5.545 m/s