newton's laws of motion

Question

a 6kg bomb at rest explodes into 3 equal pieces p,q and r . if p flies with speed 30m/s and q with speed 40m/s making an angle 90^o with direction of p, the angle between the direction of motion of p and r is about _______

Answer 1

This is a conservation of (vector) momentum problem.   It can be solved by choosing a coordinate system for the momentum vectors and then imposing the condition that the three momentum vectors sum to zero.

The easiest choice for the coordinate system is to choose the x axis to be along the direction of motion of p.   With this choice,    the momentum coordinates of p are (30 x 2, 0) and the momentum coordinates of q are (0, 40 x 2).

(The factor of 2 is there because the mass of each piece is 2kg).  The momentum coordinates of r must be

(- 30 x 2, - 40 x2)  for the three momentum vectors to sum to zero.  The endpoint of the r vector is in the third quadrant.  The reference point in the first quadrant is  (30 x 2 , 40 x 2).  The distance of this point from the origin is 50 x 2 (Pythagorean theorem).   The vector associated with this first quadrant point makes an angle with the x axis ( the direction of p).   The cosine of this angle is (30 x 2)/ (50 x 2) =  0.6.  A calculator can be used to find that this first quadrant angle is 53.13 degrees.   The angle of interest in the third quadrant is  180 - 53.13 = 126.87 degrees.

 

The 126.87 is measured clockwise from the direction of p (the x axis).  

Answer 2

the three vectors must add to zero in the x and y directions.

 

if the vector for -r makes an angle θ with p, then the angle between p and r is 180 degrees - θ

 

tan θ = 40/30 = 4/3

 

θ = 53.13

 

Therefore the angle is 180 - 53.13 ≈ 127 degrees