Final answer:
The magnitude and direction of the third piece of the explosion can be determined using the principle of conservation of momentum. To find the total mass initially, the masses of the three pieces need to be added together, assuming the third piece's velocity and direction are known.
Step-by-step explanation:
To solve for the magnitude and direction of the third piece after the explosion, we apply the principle of conservation of momentum. Since the system is stationary before the explosion, its total initial momentum is zero. After the explosion, the total momentum must still be zero because there are no external forces acting on the system. So, we take the momentum of the two known pieces and calculate the momentum of the third piece to cancel out the total momentum.
Let's denote the third piece as mass 'm' and its velocity as 'v'. The momentum of the 1kg piece moving north at 4 m/s is (1 kg)(4 m/s) north, and the momentum of the 2kg piece moving east at 20 m/s is (2 kg)(20 m/s) east. Therefore, the third piece must have momentum equal in magnitude but opposite in direction to the vector sum of these two momenta to maintain the total momentum at zero.
To find the mass of the whole object before the explosion, given that the third piece moves at 30 m/s, we can use the same conservation principles. Adding up all the masses, including the third piece, will give us the mass of the object before the explosion. However, more information, such as direction, is needed for a complete solution.