Physics Question on Relative Motion?

Question

I have a couple of questions on relative motion
1. Two swimmers can swim equally fast relative to the water. they have a race to see who can swim across a river in the least time. Swimmer A swims perpendicular to the current and lands on the far shore downstream, because of the current. Swimmer B swims upstream at an angle to the current and lands on the shore directly opposite the starting point. Wo crosses the river in the least time. Explain your answer.

2. A ball is whirled on the end of a string in a horizontal circle of radius R and period T. If the period is increased by a facotr of 2 (keeping the radius constant), what speed must you have in the comparison to the original speed to maintain the same centripetal acceleration as before.

Any help will be much appreciated.

Answer 1

Since the object is to cross the river and downstream displacement is not a problem, then you want to maximize your perpendicular speed.

The perpendicular swimmer has
x(t)=v*t
and the swimmer at an angle, say th, has
x(t)=v*cos(th)*t

L=v*t1 for the first
and
L=v*cos(th)*t2 for the second
t1/t2=cos(th), which is always less than 1, therefore, t1
2) If the period is increased by a factor of 2, then w2=w1/2
since T=2*pi/w

centripetal acceleration is
-w^2*r
since the radius is constant, the speed and the period are related, you have to vary the radius for the speed to change and have the same centripetal acceleration.

j

Answer 2

If th breeze is 3 m/s then th mosquito desires to go -3 m/s and vice versa. you have a adverse velocity because of the fact velocity is velocity with course and, for sure, -3+3=0