Question

Two cannons are each loaded with one cannonball. One cannonball has twice the mass, but is fired from has a barrel that is twice the length of the other. Which cannonball, if either, will travel farther? Justify your answer with any necessary equations. Assume each cannon applies the same force, and assume that force is constant until the cannonball exits the barrel.

Answer 1

Solution:

Let us go to the basics first.

The cannonball with higher velocity at exit, should travel farther.

Let us check in which case this is possible.

Case1:

One cannonball has twice the mass, but is fired from a barrel that is twice the length of the other.

M = 2m

L = 2l [where, m = mass of other ball and l = length of other barrel]

F = F = constant (same as given in question)

From Newton's eqn. of motion, we know that:

V² = U² + 2aL [where, V = Exit velocity at the exit of barrel; U = initial velocity = 0 since cannonball will be at rest initially; a = acceleration]

=>V² = 2aL ...............Eqn.1

Now, F = Ma

=> a = F/M

Thus, eqn.1 becomes:

V² = 2LF/M ..................Eqn.2

For lighter cannonball adopting the same approach and equations as mentioned above, We can write eqn.2 directly:

v² = 2lF/m ....Eqn3 (For second cannon ball)

Eqn.2 divided vy Eqn.3:

V² / v² = (2LF / M) / (2lF / m)

=>V² / v² = (L / M) / (l / m)

=>V² / v² = (2l / 2m) / (l / m)

=>V² / v² = 1

Thus, both cannon balls will have same exit velocity and thus both will travel same.

Thanks!!!