a boat heading due north crosses a wide river with speed of 12km/h relative to the water the water in the river has a uniform speed of 5 km/h due east relative to the earth.determine the velocity of the boat relative to an observer standing on either bank and the direction of boat
Do vector addition of the two velocities to obtain boat's velocity V w.r.t.river bank:
V = sqrt(12^2+5^2) = 13 Km/hr
and
tan theta = 5/12; (theta - angle between V and north)
The boat is heading due north as it crosses a wide river with a velocity of 10 km/h relative to water. The river has a maintain velocity of 5 km/h due east. Determine the velocity of the boat with respect to an observer on the riverbank.
Positive
To determine the velocity of the boat relative to an observer standing on either bank, we need to consider the vector addition of the boat's velocity relative to the water and the velocity of the water.
Let's break down the problem step by step:
1. Start with the boat's velocity relative to the water, which is given as 12 km/h due north. We can represent this velocity as a vector pointing north.
2. Now, consider the velocity of the water, which is given as 5 km/h due east. Since we need to find the boat's velocity relative to an observer on the bank, we need to convert the water's velocity from east to north. This can be done using basic trigonometry.
The magnitude of the water's velocity in the north direction can be found using the Pythagorean theorem:
(water's velocity in north direction)^2 = (water's velocity in east direction)^2 + (water's velocity in north direction)^2
Substituting the given values:
(water's velocity in north direction)^2 = 5^2 + 0^2
(water's velocity in north direction)^2 = 25
water's velocity in north direction = √25 = 5 km/h
So, the velocity of the water relative to the observer standing on the bank is 5 km/h due north.
3. Now, we can add the boat's velocity relative to the water (12 km/h due north) and the velocity of the water (5 km/h due north) to find the boat's velocity relative to an observer standing on the bank.
Since the boat's velocity and the velocity of the water are both due north, their magnitudes can be simply added:
boat's velocity relative to the observer = boat's velocity relative to water + velocity of the water
boat's velocity relative to the observer = 12 km/h + 5 km/h = 17 km/h
4. Finally, the direction of the boat's velocity relative to the observer can be determined by considering the angle between the observer and the velocity vector. In this case, the boat is heading due north and the observer is on the bank, so the direction of the velocity vector is simply north.
Therefore, the velocity of the boat relative to an observer standing on either bank is 17 km/h due north.