# Is it theoretically possible for the orientation angle of a projectile to remain exactly equal to the orientation of velocity?

This question is sparked by my answer to this question: Is this simulation following real physics?

After examining the math, I don't see how it is theoretically possible for the situation simulated to happen.

The situation in question is whether or not a spear in projectile motion can stay oriented over the entire path such that the angle is equal to the angle of the velocity vector.

I start with proposing that the rate of change of the angle of the velocity vector must be a constant - there can be no torque applied during the projectile motion, only the initial angular velocity imparted at the release.

So taking a projectile, with say an initial velocity of <20, 20> - this would be at a 45 degree angle, the case where the condition intuitively seems most likely:

$\frac{dy}{dt} = -9.8t + 20$, $\frac{dx}{dt} = 20$

We find $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{-9.8t + 20}{20}$

And an expression for the angle, then:

$\theta = \arctan{\frac{dy}{dx}}$

Graphing the derivative of this function over the time span of the projectile shows that it is NOT a constant. This implies that there would have to be some sort of torque being applied mid-air if the angular displacement is directly proportional to $\theta$. However, it IS a very shallow bowl.

First of all, is this analysis correct?

Second, if this implies that the condition is impossible, what happens when one throws a javelin or spear perfectly? Why is it so natural to do so?