Atrebuchet was a hurling machine built to attack the walls of a castle under siege. a large stone could be hurled against a wall to break apart the wall. the machine was not placed near the wall because then arrows could reach it from the castle wall. instead, it was positioned so that the stone hit the wall during the second half of its flight. suppose a stone is launched with a speed of v0 = 25.0 m/s and at an angle of θ0 = 41.0°. what is the speed of the stone if it hits the wall (a) just as it reaches the top of its parabolic path and (b) when it has descended to half that height? (c) as a percentage, how much faster is it moving in part (b) than in part (a)?

(a) 18.9 m/s

The motion of the stone consists of two independent motions:

- A horizontal motion at constant speed

- A vertical motion with constant acceleration () downward

We can calculate the components of the initial velocity of the stone as it is launched from the ground:

The horizontal velocity remains constant, while the vertical velocity changes due to the acceleration along the vertical direction.

When the stone reaches the top of its parabolic path, the vertical velocity has became zero (because it is changing direction): so the speed of the stone is simply equal to the horizontal velocity, therefore

(b) 22.2 m/s

We can solve this part by analyzing the vertical motion only first. In fact, the vertical velocity at any height h during the motion is given by

(1)

where

is the initial vertical velocity

is the vertical velocity at height h

is the acceleration due to gravity (negative because it is downward)

At the top of the parabolic path, , so we can use the equation to find the maximum height

So, at half of the maximum height,

And so we can use again eq(1) to find the vertical velocity at h = 6.9 m:

And so, the speed of the stone at half of the maximum height is

(c) 17.4% faster

We said that the speed at the top of the trajectory (part a) is

while the speed at half of the maximum height (part b) is

So the difference is

And so, in percentage,

So, the stone in part (b) is moving 17.4% faster than in part (a).

magnityde and direction

elly's long hair:

a. if the salon only takes cash, then this isn't the problem. it doesn't matter how much she has in her checking account.   she can't get a haircut until she has the cash.

b. if she has overdraft protection, then there's no problem.   she can go ahead and write a check that her account can't cover, then sometime later, pay the bank some outrageous usury for loaning her a few bucks for a few days, but with beautiful hair.

c. if elly has some principles, and doesn't want to write a bad check, then now she has a problem.   her desire for the new hairstyle is in conflict with her honesty and integrity.

d.   if the bank will not process a check she writes, then even if she had a million $$ in her account, she couldn't use any of it.   the amount she has in her account is not the problem.   and if i may suggest, she might as well close her checking account, and put the money where it'll work for her.   it's not doing her a bit of good in a checking account, since she can't write any checks.

justin can't pay the full balance:

a. he can't even handle the credit limit he has now.   under no circumstances should he ask for a higher limit.

b.   he can ask for the grace period to be extended.   but that's no solution, the bank will almost certainly refuse anyway, and there are much better options.   he knew the rules of his credit card when he started using it.

c.   he definitely should pay the interest on the balance, plus as much of the balance as he can cover at the same time.

d.   he definitely should pay the minimum otherwise they could shut down his card.   he should also pay as much more as he possibly can.   he should also remember the awkward situation he's in, stop buying stuff on credit that he can't afford, and avoid adding any more debt.

i'd say that justin's best best option isn't on this list of choices.   what he has to do:   pay as much as he can at least the minimum payment plus as much more as he can and then avoid adding more debt.