Projectile Motion: Achieving a Semicircular Trajectory without Calculus

[JackFyre]

Hey Folks!
I've got a longish one-
At what angle must you release a projectile to achieve a semicircular trajectory(neglecting air-resistance)? Would the initial release velocity matter? and would the same criteria hold true if the value of g were different(on the moon for example)? and is there any way of proving all this without calculus?

 

[vanhees71]

Are you looking at the "near-earth approximation", where the gravitational field $\vec{g}=\text{const}$. Then you never have a semicircular trajektory, because all trajectories are either parabolae or straigt lines, as can be easily seen solving the equation of motion,
$$\ddot{\vec{x}}=\vec{g}.$$
Since $\vec{g}=\text{const}$ you just have to integrate twice with respect to $t$ and working in the intia conditions,
$$\vec{x}(t)=\vec{x}_0 + \vec{v}_0 t +\frac{1}{2} \vec{g} t^2.$$
You can of course have circular orbits around the Earth, for which
$$\vec{g}=-G m_{\text{earth}} \vec{r}/r^3,$$
where now the coordinate origin is at the center of the Earth.

This problem is of course a bit more difficult to solve, but you find it in any textbook on mechanics (just look for "Kepler problem").

 

[berkeman]

Another option would be to tie a rope to the projectile, anchor the rope some distance away, and fire the projectile straight up with some minimum velocity. That would cause the projectile to travel in a semi-circular arc, centered at the anchor point on the ground.

 

[hutchphd]

Another option would be to tie a rope to the projectile

Did you ever see the (CRAZY) dual cannon with chain between the cannonballs weapon?? Synchronization was apparently a problem!. Mow down the enemy advance...

 

[JackFyre]

Thanks. Makes it clear.