Conservation Laws and Projectile Motion

In summary: But I still don't understand why it's not working. Is there a more detailed explanation of what you're doing?In summary, the cannon fires a shell at an elevation angle, theta, and muzzle velocity, v0. The shell explodes into two fragments, with an additional energy, E, traveling in the original horizontal direction. At the highest point, the two fragments land in the ocean.
  • #1
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Homework Statement



A cannon in a fort overlooking the ocean fires a shell of mass M at an elevation angle, theta and muzzle velocity, v0. At the highest point, the shell explodes into two fragments (masses m1 + m2 = M), with an additional energy E, traveling in the original horizontal direction. Find the distance separating the two fragments when they land in the ocean. For simplicity, assume the cannon is at sea level.

Homework Equations



Conservation of Momentum:

px: Mv0cos(theta) = m1v1cos(theta) + m2v2cos(theta)

py: Mv0sin(theta) = m1v1sin(theta) + m2v2sin(theta)

Conservation of Energy

Mv0^2/2 + E = m1v1^2/2 + m2v2^2/2

Maybe

y = y0 + v0tsin(theta) – gt^2/2 and x = x0 + v0tcos(theta) at the end…


The Attempt at a Solution



First I used the kinematic equations to find the time when the two broken things will hit the ocean (they hit at the same time, I think)… t = 2v0sin(theta)/g

Then I tried to find v2 in terms of v1. I did this by squaring px and py above…

(M^2)(v0^2)(sin^2 + cos^2) = (m1^2)(v1^2) (sin^2 + cos^2) + etc.

I got v0^2 = (m1v1 + m2v2)^2/M^2 and plugged this into the conservation of energy equation…

If I can ever get my algebra correct, I was thinking about plugging this result back into the above equations and messing with things until I can get both v1 and v2 in terms of v0…

Then I can just use the x component of the kinematic equations to find their distance from the cannon, right? And take the difference?.. It seems simple enough, but its just not working out for me. Is there a better/easier way to do this? Thanks!
 
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  • #2
You are on the right track. The energy equation you have, however, is not in the most convenient form. Can you express vo in terms of the original energy (rhetorical question)?

Yeah, you have the equations down though, and the rest of the problem is just algebraic manipulation, and a lot of it at that.
 
  • #3
Thanks, Mindscrape I figured out how to write my v1 and v2 in terms of v0, so I think I can do it now.
 

FAQ: Conservation Laws and Projectile Motion

What are conservation laws and why are they important in projectile motion?

Conservation laws are fundamental principles in physics that state certain quantities, such as energy and momentum, are conserved or remain constant during a physical process. In projectile motion, these laws help to explain and predict the motion of a projectile by considering the forces acting on the object and the resulting conservation of energy and momentum.

How do conservation laws apply to projectile motion?

In projectile motion, the conservation of energy and momentum can be seen through the initial and final states of the projectile. The initial energy and momentum of the projectile are conserved throughout its flight, resulting in a predictable trajectory. This is because the only forces acting on the projectile are gravity and air resistance, which are both conservative forces and do not change the total energy or momentum of the projectile.

What is the difference between elastic and inelastic collisions in terms of conservation laws?

Elastic collisions are those in which both energy and momentum are conserved, meaning the total kinetic energy and total momentum of the system remains constant. In inelastic collisions, on the other hand, only momentum is conserved, while some energy is lost due to the deformation of the colliding objects or the conversion of kinetic energy into other forms, such as heat or sound.

How does air resistance affect conservation laws in projectile motion?

In most cases, air resistance is considered a negligible force in projectile motion, as it does not significantly affect the conservation of energy and momentum. However, in situations where the projectile is traveling at high speeds or over long distances, air resistance can play a larger role and may need to be taken into account in the calculations.

Can conservation laws be violated in projectile motion?

No, conservation laws are fundamental principles in physics and cannot be violated. However, in some cases, it may seem like these laws are being broken due to external factors, such as friction or air resistance, that were not initially considered in the analysis. In reality, the laws are still being upheld, but the additional forces are altering the final outcome of the projectile's motion.

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