An object of mass 180 g initially at rest explodes into three pieces. One piece has mass 70 g and moves directly west at speed 14 m/s, and another piece has mass 50 g and moves directly south at speed 18 m/s. Find the speed and direction of the third piece.

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.

50 inches should be the resultant vector

the total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)