Why do hailstones hurt more than raindrops when they hit you?

[planauts]

Homework Statement

In a particular thunderstorm, the hailstones and raindrops have the same mass and terminal velocity. Explain with reference to Newton's second law why the hailstones hurt more than raindrops when they hit you.

Homework Equations

$$F = ma$$

$$F = m \frac{v}{t}$$

$$F = \frac{p}{t}$$

The Attempt at a Solution

Net Force on an object equals mass x acceleration. Net Forces on both the raindrop and hailstones are different. The force of gravity is the same downward. But the force upward caused by wind resistance is different based on aerodynamic shape. Since the raindrop is not as aerodynamic as the hailstone the wind resistance will be greater, and hence acts a brake against gravity. The actual velocity of the hailstone will be greater when it hits your face. Terminal velocity is the same but neither if them ever reach.I don't think that is right. The book said:

Hint: $$F = \frac{p}{t}$$

I am not sure.

Thanks,

 

[eczeno]

the hint is referring to the impulse-momentum theorem, are you familiar with that?

 

[planauts]

Well I know that Impulse is the change in momentum.

[URL]http://upload.wikimedia.org/math/f/5/c/f5c3f3208242cb2f29114f55c7c5897c.png[/URL]

But I don't understand the explanation :-/

 

[eczeno]

well, you are misunderstanding the question a bit. if the terminal velocity of each is the same, that means they will hit you with the same speed. and if they have the same mass then they will have equal momentum.

 

[planauts]

well, you are misunderstanding the question a bit. if the terminal velocity of each is the same, that means they will hit you with the same speed. and if they have the same mass then they will have equal momentum.

Hey sorry for the VERY late reply.

If the momentum is the same, it doesn't explain why the hail hits you harder.
Would the change in momentum (impulse) of the hail be greater?

 

[Peter G.]

If they have the same momentum when they hit you and the same momentum after, their velocity will probably change to 0 m/s, they have the same change in momentum. Do you agree with that?

You want to find the force, so that's an unknown. So the last variable you're left is the one that's going to change, what do you expect?

The situation is analogous to kicking a wall and kicking a cushion with the same initial and final velocity of your leg. Why does one hurt more?

Hope that helps.

 

[planauts]

I/t = F

So the time at which the momentum changes to zero will be quicker for the hailstone than the raindrop?

Because of this, we experience a greater force?

 

[Peter G.]

Yes, exactly. The time it takes for the change of momentum to take place will determine the force in this case.

When you kick the cushion as I said, isn't your feet in contact with its surface for much for time then when you kick the wall?

Similarly, that's why in cars, for crash tests, they try and increase as much as possible the crumple zone so that the car's crash takes longer.

 

[planauts]

Great, thanks for your help! It makes more sense now.

 

[Peter G.]

No problem