A man is rowing at 8 km/h in a river 1.5 km wide in which the current is 5 km/h. (a) In what direction should he head in order to get across the river in the shortest possible time? (b) How much time will he take if he goes in this direction? (c) How far downstream will the boat have gone when it reaches the opposite side?
If you just want to get to the other side and do not care how far downstream you land, head straight across. That maximizes your speed component perpendicular to the current.
1.5 km * 1 hour/ 8 km = (1.5/8) hours = 11.25 minutes
(5 km/hour)(1.5 /8)hours = 0.9375 km
a. Vr = Vb + Vc = 8 = Resultant velocity.
Vb + (-5i) = 8,
Vb = 8 + 5i = 9.43km/h[32o] N. of E. = Velocity and heading of boat.
b. d = V * T = 1.5.
8T = 1.5,
T = 0.1875 h.
c. Since the affect of the current is offset by the 32o heading, the boat goes directly across the river.
Henry, it said get across as fast as possible. It did not say to go to a point directly across. Your across component is maximum if you ignore the current and head straight across, drifting downstream of course.
To find the optimal direction and the time it takes for the boat to cross the river, we can break down the problem into its components.
(a) In what direction should he head in order to get across the river in the shortest possible time?
To determine the shortest time, the boat needs to minimize the effect of the current. Considering the width of the river and the speed of the current, it is apparent that the boat will be pushed downstream while trying to cross. Therefore, the boat should head upstream at an angle that compensates for the downstream drift caused by the current.
(b) How much time will he take if he goes in this direction?
To calculate the time, we can use the concept of relative velocity. The boat's speed in still water is 8 km/h, and the current's speed is 5 km/h. By subtracting the downstream component of the current (5 km/h) from the boat's speed (8 km/h), we get the effective speed of the boat moving perpendicular to the current.
Effective speed = Boat's speed - Downstream component of current
Effective speed = 8 km/h - 5 km/h
Effective speed = 3 km/h
The effective speed of the boat is now 3 km/h. To calculate the time it takes to cross the river, we divide the width of the river by the effective speed.
Time = Distance / Speed
Time = 1.5 km / 3 km/h
Time = 0.5 hours = 30 minutes
Therefore, it will take the boat 30 minutes to cross the river when heading upstream in the optimal direction.
(c) How far downstream will the boat have gone when it reaches the opposite side?
To determine how far downstream the boat will have gone, we need to calculate the downstream component of the current's speed over the crossing time. Since the crossing time is 30 minutes (0.5 hours), we can calculate the downstream distance using the equation:
Downstream distance = Downstream speed × Time
Downstream distance = 5 km/h × 0.5 hours
Downstream distance = 2.5 km
Therefore, when the boat reaches the opposite side of the river, it would have drifted downstream by approximately 2.5 km.