Throwing Balloons

If $w$ is half of Jim's width, $d$ is the distance of Jim to the balloon, $u$ is the speed of the bus, then one could launch the balloon at speed $v = \frac{d}{w} u$ and still hit Jim. Realistic values would be $d = 5$ m, $w = 0.3$ m, $u = 6$ m/s, so that $v = 100$ m/s or 225 mph-- a very fast balloon.

But the balloon must have a decent speed, otherwise it splashes on the ground before reaching Jim. If the balloon is tossed from height $h$ , we need a minimum speed of $v = d\sqrt{g/2h}$ . With $h = 1.5$ m and $g = 10$ m/s/s, this minimum speed is nearly 10 m/s, or 23 mph-- still pretty fast.

And how big is jim...

This problem is only difficult insofar as the language used. Colloquially if you throw an object at someone that doesn't mean you are throwing the object at the stationary point they currently occupy in your reference frame.

Otherwise none would ever catch a pass in football. This isn't even including gravity which will further throw you off

The qualifier is that they are point particles - otherwise the question is not really solvable for the real world without more data. Although of course in the real world the human brain will allow for speed angle etc.

As also the fact Sam will probably be friendless!

Actually, there is also another reason the balloon will miss Jim. Being a bus, the window will be up. No balloon will be thrown out.

no. excses throw a bus throwthly in the days off ouer,s.thank uou!

While I understand that we're looking at the problem from a maths/physics perspective, in reality, the balloon is very likely to hit Jim because the human brain is quite good at correcting speed differentials and inertia on the go. A simple way of corroborating this is to walk briskly while tossing a ball high into the air. By all means, the ball should always fall behind you, more or less onto the spot you were when you toss it, but you'll find out that with the slightest application of intentional direction, you can always make the ball fall back into your hand, your brain effortlessly calculating and executing what on a paper would be a complex parabolic trajectory.

It bears mentioning, though, that if the bus is moving at a very high speed, the balloon is likely to miss if only because the thrower's arm is not capable of moving fast enough to compensate the distance traversed by the bus in that short lapse.