Answer: the speed is 22.17m/s, and the direction is 42.56° from north to east.
Explanation:
The mass of the object is 180g, and it initialy has a velocity of 0m/s, so the total momentum is P = 0
In this problem we must use the conservation of the momentum, this means that the final momentum must be equal than the initial momentum.
we have 3 pieces:
One has a mass of m1 = 70g and v1 = 14m/s west.
other has a mass of m2 = 50g and v2 = 18m/s south.
this two pieces have a momentum of:
p1 = 70*14 (g*m/s) west = 980 (g*m/s) west
p2 = 50*18 (g*m/s) south = 900 (g*m/s) south.
now, the total momentum must be zero, so P3 = 980 (g*m/s) east + 900 ( (g*m/s) north.
first, knowing that the total mass is 180g, we have that the mass of the third piece is:
m3 = 180g - 70g - 50g = 60g
now, for the momentum we will have that one component goes north and other east, if we define the angle A as the angle that goes from north to east (such that A = 0° means that we are moving directly to north) we will have:
p3 = 60g*v*(cos(A)) to north + 60g*v*sin(A)
then we have a system of equations:
60*v*cos(A) = 980
60*v*sin(A) = 900
Here we can use the quotient of equation 2 and equation 1, in this way we remove the variable v and we can solve it for A:
(60*v*sin(A) )/(60*v*cos(A)) = 900/980
tan(A) = 900/980
Atan(tan(A)) = A = Atan(900/980) = 42.56°
now with this angle we can find the value of v:
60*v*cos(42.56°) = 980
v = 980/(60*cos(42.56°) = 22.17 m/s
then the speed is 22.17m/s, and the direction is 42.56° from north to east.
