Suppose a particle is projected initially at speed $u$ downwards towards the ground at an angle $\alpha$ beneath the horizontal at a point $P$ above the ground. Let $O$ be a point on the ground, vertically below $P$. The particle will hits the ground at some point $R$.

If we throw the particle at the same speed $u$ towards the ground, but this time at a larger angle $\beta$, $\beta > \alpha$. The particle now hits the ground at some point $T$.

Is the horizontal distance $OT$ greater than or less than the horizontal distance $OR$?

I think the answer to this is that $OT < OR$.

However, this is from intuitive reasoning only: if the particle is thrown downwards at a larger angle, I feel like it is aimed more directly towards the ground and hence the distance $OT$ will be smaller than $OR$. It is hard to explain my intuition, but I'm pretty sure the distance is smaller.

Explaining this physically is difficult.

I tried to think that the range of a projectile equals $$u\cos \alpha \times t$$ where $t$ is the time of flight. If we increase $\alpha$ to $\beta$ (of course only up till $90^o$, which is the physical limit anyways), then this decreases for $t$ fixed. However, I don't think $t$ is fixed, so this is probably not the right explanation.

IS there any simple explanation for this problem? I feel like this is a 'simple' thought experiment and there is likely to be a simple qualitative explanation.


You are correct in guessing that $OT < OR$, and to solidify this intuition, there are only two factors to consider -

1) Since you increased the angle with which you hurl the object downward, the horizontal component of the velocity is shortened ($u \cos \theta$ gets smaller).

2) Also, because of the increase in the angle, the vertical component gets larger.

These two help shorten the range in unison. How? Well, the horizontal range is given by the formula $$ ut\cos \theta,$$ So the horizontal range can reduce either because of a reduction in $u\cos \theta$ or time $t$ or both together.

What's interesting is because the downward velocity increased as you increased the angle, the time it will take for the particle to reach the ground also decreases

As a result, you have reduced both $t$ and $u\cos\theta$ by increasing the angle. So it must be true that $\boxed {OT < OR}$.

Hope this helps!


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