a man can swim with a speed of 4 km per hr in still water. How long does he take to cross a river 1 km wide if the river flows steadily at 3 km per hr and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?
Ans: the speed of the swimmer will be approx 1 km
Are you studying vectors?
the resulting velocity vector is (4,0) + (0,3)
= (4,3)
magnitude of vector = √(4^2+3^2)
= √25 = 5
time to go 1 km at 5 km/h
= 1/5 hours or 12 minutes
angle between the resultant an normal:
tanØ = 3/4
Ø = 36.87°
for distance down the river where he lands:
tanØ = x/1
x = tanØ = .75
To solve this problem, we need to consider the swimmer's speed in still water and the speed of the river current. Since the swimmer is making his strokes normal to the river current, the current's speed will affect the swimmer's velocity.
Step 1: Calculate the swimmer's effective speed
The effective speed is the vector sum of the swimmer's speed in still water and the speed of the river current. We can use the Pythagorean theorem to find the effective speed:
effective speed = √((speed in still water)^2 + (river current speed)^2)
= √((4 km/hr)^2 + (3 km/hr)^2)
= √(16 km^2/hr^2 + 9 km^2/hr^2)
= √(25 km^2/hr^2)
= 5 km/hr
Step 2: Calculate the time taken to cross the river
Since the swimmer is crossing a river that is 1 km wide, we can use the formula:
time = distance / speed
time = 1 km / 5 km/hr
time = 0.2 hr
Step 3: Calculate the distance downstream
To find how far downstream the swimmer has gone when reaching the other bank, we multiply the speed of the river current by the time taken to cross the river:
distance downstream = river current speed * time
distance downstream = 3 km/hr * 0.2 hr
distance downstream = 0.6 km
Therefore, the swimmer takes 0.2 hours (or 12 minutes) to cross the river, and he travels approximately 0.6 km downstream when he reaches the other bank.