A plate falls vertically to the floor and breaks up into three pieces, which slide along the floor. Immediately after the impact, a 320-g piece moves along the x-axis with a speed of 2.00 m/s and a 355-g piece moves along the y-axis with a speed of 1.50 m/s. The third piece has a mass of 100 g. In what direction does the third piece move?You can neglect any horizontal during the crash.
Considering the system of coordinates with (x,y) axis in the floor plane, the fall of the plate is initially directed towards the negative direction of z axis. Touching the floor the force on the plate is normal on it, thus on the positive direction of z axis and cancels the initial (-z) direction momentum. Since there is no force on the plate on the (x,y) plane, the total impulse on the (x,y) plane need to be the same before and after the collision. Therefore written as vectors one has
`m_1*v_x +m_2*v_y +m_3*v_3 =0`
`|v_3| = sqrt((m_1*v_x)^2+ (m_2*v_y)^2)/m_3`
`|v_3| =sqrt((0.32*2)^2 +(0.355*1.5)^2)/0.1 =8.326 m/s`
The angle that `v_3` is making with the positive direction of the x axis is
`theta = 180 +arctan ((m_2*v_y)/(m_1*v_x)) = 180+arctan((0.355*1.5)/(0.32*2)) =180+39.76 =219.76 degree`
The figure is below.
Answer: the third piece moves in the (x,y) plane making an agle of 219.76 degree with the positive direction of x axis.