Projectile Explosion: Finding the Larger Mass's Landing Spot

[h.a.y.l.e.y]

I have a problem where I am told a projectile explodes at a distance L (in the x direction) into 2 pieces of masses 1/4 and 3/4. The smaller mass lands back at the origin. Where does the larger mass land?

I have tried CofM arguments but think I need to find the time it takes to reach the top of its trajectory in order to find how long it takes for the masses to land. By the way, I am not told the angle it is fired at.
Could someone enlighten me further please...

 

[OlderDan]

I have a problem where I am told a projectile explodes at a distance L (in the x direction) into 2 pieces of masses 1/4 and 3/4. The smaller mass lands back at the origin. Where does the larger mass land?

I have tried CofM arguments but think I need to find the time it takes to reach the top of its trajectory in order to find how long it takes for the masses to land. By the way, I am not told the angle it is fired at.
Could someone enlighten me further please...

If you take advantage of the symmetry of projectile motion, conservation of momentum will tell you the change in velocity of both objects at the point of explosion in terms of the velocity just before the explosion . If you write the initial velocity algebraically, you can express the vertical height and the velocity in terms of horizontal postion, x, and initial velocity components, eliminating time from the equations. You are correct that to get a numerical result you would need more specific information, but you can get an algebraic result for the time of flight after the explosion in terms of the initial velocity components and L. Using that time, and the post-explosing horizontal velocity you can calculate the landing point of the larger mass. The algebra gets a bit involved, but there is nothing worse than solving a quadratic equation.

 

[whozum]

With no external forces acting, momentum is conserved:

$$m\vec{v_i} = m\vec{v_f}$$

What Dan's saying is that the center of mass' momentum will continue to travel in the same direction, so if it breaks into two pieces,

$$m_1\vec{v_i} + m_2\vec{v_i} = m_1\vec{v_f} + m_2\vec{v_f}$$

In the above equation, you know the left hand side, and th first term on the right hand side.