- #1

TheGreatDeadOne

- 22

- 0

- Homework Statement
- A projectile of mass m is fired with initial velocity of module v_0 at an elevation angle of 45◦. The projectile explodes in the air in two pieces of masses m/3 and 2m/3. The pieces continue to move in the same plane as the entire projectile and reach the ground together. The smaller piece falls at a distance of 3(v_0)^2 /2g from the launch point. Determine the range of the largest chunk. Neglect air resistance

- Relevant Equations
- ..

This problem I already solved using another resource (just get the coordinate of the center of mass reach and from it, get to the larger mass. R = (3v02) / (4g)). But I'm having some trouble calculating using moment conservation. Here what I've done so far:

$$ 3\vec v_0 = \vec v_1 +2\vec v_2 $$

As the fragments fall in the same time interval, the vertical components of their velocities are the same, since in the act of the explosion, they depart from the same height:

$$ 3(\cos{\theta},\sin{\theta})= (v_{1,x},v_{1,y}) + 2(v_{2,x},v_{2,y})

=(v_{1,x},v_{1,y}) + 2(v_{2,x},v_{1,y})$$

$$ \therefore v_0 \sin{\theta}=v_{1,y}$$

In addition, we can equalize the fall time intervals of each fragment using the horizontal component of each fragment (uniform movement):

$$ (v_{1,x},v_{1,y})= (v_0\cos{\theta},v_{0}\sin{\theta})$$

$$ (v_{2,x},v_{2,y})= (2v_0\cos{\theta},2v_{0}\sin{\theta})$$

Range:

$$t_1=t_2$$

$$\frac{R_1}{v_{1,x}}=\frac{R_2}{v_{2,x}}$$

$$R_2=\frac{R_1 v_{2,x}}{v_{1,x}}=\frac{3v_0^2}{g}$$

$$ 3\vec v_0 = \vec v_1 +2\vec v_2 $$

As the fragments fall in the same time interval, the vertical components of their velocities are the same, since in the act of the explosion, they depart from the same height:

$$ 3(\cos{\theta},\sin{\theta})= (v_{1,x},v_{1,y}) + 2(v_{2,x},v_{2,y})

=(v_{1,x},v_{1,y}) + 2(v_{2,x},v_{1,y})$$

$$ \therefore v_0 \sin{\theta}=v_{1,y}$$

In addition, we can equalize the fall time intervals of each fragment using the horizontal component of each fragment (uniform movement):

$$ (v_{1,x},v_{1,y})= (v_0\cos{\theta},v_{0}\sin{\theta})$$

$$ (v_{2,x},v_{2,y})= (2v_0\cos{\theta},2v_{0}\sin{\theta})$$

Range:

$$t_1=t_2$$

$$\frac{R_1}{v_{1,x}}=\frac{R_2}{v_{2,x}}$$

$$R_2=\frac{R_1 v_{2,x}}{v_{1,x}}=\frac{3v_0^2}{g}$$

Last edited: