1. The problem statement, all variables and given/known data An object is launched from the origin. At the peak of it's trajectory, it explodes into two fragments, one of mass m and the other of mass 3m. The smaller fragment of mass m lands back at the trajectory. How far from the origin does the second fragment land? 2. Relevant equations Conservation of Momentum: P = mv Conservation of Energy: KE = .5mv^2 only because explosion happens at singular height, so GPE irrelevant Center of Mass: 1/M Ʃ mx Kinematic equations 3. The attempt at a solution The small mass m is at the origin, and the large mass 3m is some distance away. I will call this distance x. Using the Center of Mass formula, (m(0)+3m(x))/(3m+m) = 3/4 x. So we know that the center of mass is 3/4 x. We also know that if the projectile did not explode, it would land at this spot. Now, there are no times or velocities given. Using the kinematic equations and considering the even of the original projectile launching from the ground to the top of it's trajectory, we can say that: x0=0 xf= 3/8 x ax=0 ay=g y0=0 This info combined with different kinematic equations gives these expressions: 0 = voy^2 +2gh voy = gt h = gt^2 + .5gt^2 = 3/2gt^2 0=g^2 t^ + 2gh -g^2 t^4 / 2g = (3gt^2)/2 t^2 = 3 t = sqrt (3) This is the time it takes for our projectile to reach the top of it's path. Now, I am ultimately out to get the x distance where the large mass is laying. I think that I should continue manipulating the kinematic equations, but I am not sure that I have enough information. I could fiddle around and get velocities for each of the pieces, I think. I could plug them into the conservation of momentum equation: but this wouldn't do me any good with finding a distance! Likewise with the conservation of energy. Another useful piece of info: if m1 = 3m2, then according to cons. of momentum, I'm pretty sure that 3v1= v2. Basically, I am not sure which direction to go in.... I've been up and down this problem for a long time already, and I'm hoping that I am just forgetting some obvious point! Thanks in advance!