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 forces during the crash.
A) 216.9° from the x-axis
B) 219.8° from the x-axis
C) 36.9° from the x-axis
D) 39.9° from the x-axis
E) 39.8° from the x-axis

Question

Quizplus · Accepted Answer

To find the direction of the third piece, we use the principle of conservation of momentum. The total momentum before the impact (when the plate was falling vertically) in the horizontal plane was zero, so it must remain zero after the impact. The momentum of the first piece is $320\,g \times 2.00\,m/s = 640\,g\cdot m/s$ along the x-axis, and the momentum of the second piece is $355\,g \times 1.50\,m/s = 532.5\,g\cdot m/s$ along the y-axis. The third piece must have a momentum that exactly cancels out these to maintain the total momentum at zero. Using vector addition, the angle $\theta$ of the third piece's velocity vector can be found using the arctan of the ratio of the y-component to the x-component of the momentum. Thus, $\theta = \arctan\left(\frac{532.5}{640}\right)$, which gives an angle from the x-axis. However, since the third piece's momentum is in the opposite direction to balance the other two pieces, we must add 180° to the calculated angle to get the correct direction from the positive x-axis, leading to an angle close to 219.8°.

A Plate Falls Vertically to the Floor and Breaks Up