Explosions in two dimensions: 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.

Question

Quizplus · Accepted Answer

To find the direction of the third piece, we use the conservation of momentum. The total momentum before the collision (when the plate was falling vertically) in the horizontal directions (x and y) was zero. Thus, after the collision, the total momentum in these directions must also be zero. The momentum of the third piece must counterbalance the momentum of the other two pieces. For the x-direction, the momentum of the 320-g piece is $0.320 \, \text{kg} \times 2.00 \, \text{m/s} = 0.64 \, \text{kg} \cdot \text{m/s}$. There's no initial x-direction momentum for the other two pieces.For the y-direction, the momentum of the 355-g piece is $0.355 \, \text{kg} \times 1.50 \, \text{m/s} = 0.5325 \, \text{kg} \cdot \text{m/s}$. There's no initial y-direction momentum for the other two pieces.The third piece must have equal and opposite momentum to make the total momentum in both x and y directions zero. Its x-momentum is $-0.64 \, \text{kg} \cdot \text{m/s}$ and its y-momentum is $-0.5325 \, \text{kg} \cdot \text{m/s}$.To find the direction, we calculate the angle $\theta$ from the x-axis using the arctan of the ratio of y-momentum to x-momentum: $\theta = \arctan\left(\frac{0.5325}{0.64}\right)$.This calculation gives an angle, but since both momenta are in the negative direction (third quadrant), we adjust the angle to the correct quadrant by adding 180°, resulting in an angle of approximately 219.8° from the x-axis.

Explosions in Two Dimensions: a Plate Falls Vertically to the Floor