A man wished to travel due north in order to cross river 5kilometer wide flowing due east at 2kilometer per hour. If he can roll at 9kilometer per hour in still water, find 1. The direction in which he must head his boat in order to get to his destination directly opposite to his starting point. 2. The resultant velocity of the boat in the river. 3. The time taken to cross the river.
you have a right triangle, with the hypotenuse = 9 (his actual speed), and one leg=2 (the current)
so, his resultant velocity must be upstream at an angle x such that
sin(x) = 2/9
and the actual resultant speed s (due north) is √(9^2-2^2)
Having gotten his speed s, the time needed is 5/s
1. 2 + 9i = 9.22km/h[77.5o] N. of E. = 12.5o E. of N.
Direction = 12.5o W. of N.
2. Vr = 9.22 km/h(part 1).
3. V*T = d = 5km.
9*T = 5, T = 0.555 h.
1. To get to his destination directly opposite his starting point, the man must head his boat in a direction that combines the speed of the river current and the speed of his boat. Since the river is flowing due east, he needs to head his boat slightly towards the north-west.
2. The resultant velocity of the boat in the river can be determined using vector addition. The boat's speed in still water is 9 kilometers per hour, and the river current is flowing at 2 kilometers per hour due east. Using the Pythagorean theorem, the resultant velocity can be calculated as follows:
Resultant Velocity = sqrt((9^2) + (2^2))
Resultant Velocity = sqrt(81 + 4)
Resultant Velocity = sqrt(85) kilometers per hour
Therefore, the resultant velocity of the boat in the river is approximately 9.22 kilometers per hour.
3. The time taken to cross the river can be calculated by dividing the width of the river by the resultant velocity of the boat. In this case:
Time taken to cross the river = Width of the river / Resultant Velocity
Time taken to cross the river = 5 kilometers / 9.22 kilometers per hour
Time taken to cross the river = 0.54 hours or approximately 32.4 minutes
Therefore, it will take approximately 0.54 hours or 32.4 minutes to cross the river.
To answer these questions, we need to break down the problem into two components: the man's velocity in still water and the velocity of the river.
1. Direction:
To reach his destination directly opposite his starting point, the man's boat must counteract the river's current. Since the river is flowing due east, the man's boat must head in a direction that is perpendicular to the current. In other words, he needs to head northeast (45 degrees) in order to cancel out the eastward flow of the river.
2. Resultant Velocity:
The resultant velocity of the boat in the river is the vector sum of the man's velocity in still water and the velocity of the river. In this case, the man's velocity in still water is 9 km/hr north and the velocity of the river is 2 km/hr east.
To find the resultant velocity, we can use vector addition. We can treat the velocities as vectors and add them using the Pythagorean theorem:
Resultant velocity = √((9 km/hr)^2 + (2 km/hr)^2)
Calculating this, we get:
Resultant velocity = √(81 km^2/hr^2 + 4 km^2/hr^2) = √85 km^2/hr^2
Therefore, the resultant velocity of the boat in the river is approximately √85 km/hr.
3. Time taken to cross the river:
The time taken to cross the river can be calculated by dividing the distance of the river by the horizontal component of the resultant velocity.
Distance of the river = 5 km
Horizontal component of the resultant velocity = velocity of the river = 2 km/hr
Time taken to cross the river = Distance of the river / Horizontal component of the resultant velocity
Time taken to cross the river = 5 km / 2 km/hr = 2.5 hours
Therefore, the time taken to cross the river is 2.5 hours.