A boat must head due north across a river channel that has a current of 9 km/hr towards the west. The boat is capable of traveling at 28 km/hr in still water. If the boat heads into the current to keep its true north course, what will be the resultant velocity of the boat?
I'm really having trouble with this problem. After finding the resultant, you have to find how long it will take to cross.
Vc + Vb = Vbc.
=9 + Vb = 28i.
Vb = 9 + 28i = 29.4km/h[72.2o] N. of E. = 17.8o E. of N.
To find the resultant velocity of the boat, you need to consider the velocities of the boat and the current separately and then combine them.
Let's break down the problem into components:
1. The boat's velocity in still water is 28 km/hr, which we'll call its horizontal velocity (Vb-x).
2. The current's velocity is 9 km/hr towards the west, which we'll call its vertical velocity (Vc-y).
Since the boat heads due north, its course is perpendicular to the current. Therefore, there is no horizontal component of the current affecting the boat's velocity.
To find the resultant velocity:
1. The horizontal component of the boat's velocity remains unchanged because there is no horizontal component of the current affecting it. Hence, the horizontal component of the boat's velocity is still 28 km/hr (Vb-x).
2. The vertical component of the boat's velocity is affected by the current. The current's velocity, being 9 km/hr towards the west, opposes the boat's progress towards the north. So, the vertical component of the resultant velocity will be (Vb-y) - (Vc-y).
Now, let's calculate the components:
1. The horizontal component of the boat's velocity (Vb-x) remains unchanged at 28 km/hr.
2. The vertical component of the boat's velocity (Vb-y) is given by:
Vb-y = (Vb * sin θ) = (28 km/hr * sin 90°) = 28 km/hr
Since the boat heads into the current to maintain its true north course, the current's velocity needs to be subtracted from the boat's velocity during the calculation.
3. The vertical component of the current's velocity (Vc-y) is -9 km/hr because it acts in the opposite direction.
(When calculating vertical components, consider upward-positive and downward-negative conventions.)
Now, let's calculate the resultant velocity:
1. The horizontal component of the resultant velocity remains unchanged: Vr-x = Vb-x = 28 km/hr.
2. The vertical component of the resultant velocity is given by:
Vr-y = Vb-y - Vc-y = (28 km/hr) - (-9 km/hr) = 28 km/hr + 9 km/hr = 37 km/hr.
Therefore, the resultant velocity of the boat is a speed of 28 km/hr due north and a speed of 37 km/hr towards the west.
To find how long it will take to cross, you need to use the resultant velocity. First, determine the total distance to be crossed. Once you have that, you can divide it by the resultant velocity to find the time taken.
Please provide the width of the river channel, and I can assist you further with the time calculation.