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## Moving Target

This is one of the more interesting problems I've seen this year, Its got relative motion, trig, kinematics, and systems of equations.

"in an action-adventure film, the hero throws a grenade from his car moving at 'V_h' to the baddies car going at 'V_b' which starts at distance 'd' away. If the grenade is thrown at an angle 'theta', how fast does it need to be thrown 'V_0'? How far away does it land 'L' ?"

First, let's draw out a picture and name some variables. Also, convert the cars speedometer, either km/h or mph, to m/s

This problem would be just like a regular cannonball problem if the cars' velocities were the same, we can infer that subtracting them to find the relative velocity is useful. The throwing velocity also has the hero's car's velocity added to it with vector addition before being plugged into the kinematic equations.

Now we have all the equations needed to solve for V_0. By finding the initial velocity of the grenade relative to the baddie (what would this problem look like if he was moving still) the math simplifies into a regular cannonball-off-a-cliff problem.

I skipped some algebra here, but there is no way to directly solve for V_0, so it's arranged into the standard form of a quadratic and then plugged into the quadratic formula. Even more algebra is skipped below to simplify this. Email me if you find a mistake.

The plus or minus implies there are 2 solutions, but one will be negative. Usually the positive makes sense, but the negative could correspond to throwing it backward after passing the car and is probably incorrect. But if V_hb is negative (the baddie is going faster) then the other sign makes more sense, so let's leave it in the general solution.

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