A boat is to be driven from a point on the South bank of a river which flows West-to-East to a destination on the North Bank. The destination is in a direction of N60E from the starting point. The river is 0.2 km wide. The driver of the boat finds that in order to make the trip in a time of 6 min, the boat must maintain a speed of 2.5km/hr relative to water.
a) Find the component of the boat's velocity perpendicular to the bank.
b) Find the downstream component of the velocity contributed by the boat's engines.
c) Find the total downstream component of the boat's velocity.
a) (velocity component across)=
(river width)/crossing time)= 2.0 km/hr
b) If the boat's speed with respect to water is 2.5 km/h and the component across the river is 2.0 km/h, the downstream component must be 1.5 km/h (Consider the Pythagorean theorem).
c) Since it travels at a 60 degree angle to the perpendicular to the shore, the total velocity realtive to land is 2.0/cos 60 = 4.0 km/h and the total downstream velocity is 4.0 sin 60 = 3.464 km/h
1.964 km/h of that is the flow velcoity of the river
So this is someone from Pryce's class haha. who asked this?
a) To find the component of the boat's velocity perpendicular to the bank, we can use basic trigonometry. Let's call the angle between the direction N60E and the direction perpendicular to the bank as θ.
Since the destination is in a direction of N60E from the starting point, the angle θ would be 90° - 60° = 30°.
Now, to find the component of the boat's velocity perpendicular to the bank, we can use the formula:
Perpendicular Component = Total Velocity * cos(θ)
The total velocity of the boat is given as 2.5 km/hr.
So, the component of the boat's velocity perpendicular to the bank is:
Perpendicular Component = 2.5 km/hr * cos(30°)
To calculate this value, we need to convert the angle from degrees to radians, as most trigonometric functions in math libraries work with radians.
Converting 30° to radians:
θ_radians = 30° * π/180°
Now, we can calculate the perpendicular component:
Perpendicular Component = 2.5 km/hr * cos(θ_radians)
b) To find the downstream component of the velocity contributed by the boat's engines, we need to consider the effect of the river's flow.
Given that the boat must maintain a speed of 2.5 km/hr relative to water, and the river is flowing west-to-east, we can subtract the westward flow caused by the river from the boat's speed.
According to the problem statement, the river is 0.2 km wide. In 6 minutes, which is the given time, the boat must travel across the river to reach the destination on the north bank.
To calculate the downstream component of the velocity due to the engine, we use the formula:
Downstream Component = Total Velocity - River Flow Velocity
The river flow velocity can be calculated as the distance traveled across the river divided by the time taken.
Distance across the river = 0.2 km
Time taken = 6 min = 6/60 hr = 0.1 hr
River Flow Velocity = Distance across the river / Time taken
c) To find the total downstream component of the boat's velocity, we can simply add the downstream component contributed by the boat's engines to the downstream component due to the river flow.
Total Downstream Component = Downstream Component (due to boat's engine) + Downstream Component (due to river flow)