pls tell me what sign conventions have to be followed in problems of vertical motion under gravity?

also, for river boat problems, boat takes shortest time to cross the river ;-

relative velocity of boat is perndicular to the river flow

or

actual velocity is perpendicular to the river flow?

For gravity problems ... best sign convention : up : +ve and down : -ve.

Consider the problem:

A river flows west to east at a speed of 5m per minute. A man on the south bank can swim at the rate of 10m per minute in still water. In which direction should he swim so as to cross the river 1) in the smallest possible time 2) along the shortest path

Solution

Take the normal NSEW directions (N up, E right, etc). Assume XY axes such that+ve X is along east and +Y along north. Let the origin of the coordinate system be at a point on the south shore from where the man starts.

Speed of man wrt water, v_MW = 10 m/min

Velocity of water wrt to earth, v_WE = 5 i m/min

Let velocity of man wrt eath be v_ME.

Let the width of the river be d.

(a) Let the man try to swim in a direction making with the X axis.

v_MW = 10(cos i + sin j)

v_ME = v_MW + v_WE  = (10cos + 5) i + 10sin j

Time taken by the man to cross, T = d/10sin

T is minimum if sin is maximum, ie, = /2

Hence the man should swim perpendicular to the current if he is to reach in shortest possible time.

(b) The shortets possible path is pependicular to the river. For the man to follow this path, v_ME must not have any velocity component along X axis.

10cos + 5 = 0

= 120⁰

Hence the man should swim in a direction making 120⁰ the +ve X direction