A boat has a speed of 15ms-1 in still water. It is used to cross a river 500m wide along which there is a strong current of 0.9ms-1. The boat is being directed to the opposite bank, but is carried downstream.
Calculate:
a) The boat velocity relative to the bank
b) The distance the boat is carried downstream
c) The time taken to cross
d) The direction the boat must take, relative to the water if it is to cross directly
e) The time taken to cross directly, if it took the direction you calculated in part b.
Solutions would be a massive help guys. Cheers.
x velocity = 15 across
y velocity = .9 downstream
note order of answers
c. time to cross = 500/15 = 33.33 seconds
b. downstream y = .9 * 33.33 = 30 meters
a. sqrt (.9^2+15^2)
angle to x axis = A = tan^-1(.9/15)
d. sin^-1 (.9/15) toward upstream
e. component of speed across = 15 cos(angle from part d)
then
time = 500/that component
To solve this problem, we can use the concepts of vector addition and relative motion. Let's break down each part of the problem and calculate the answers step-by-step:
a) The boat velocity relative to the bank:
- The boat has a speed of 15 m/s in still water.
- The current of the river has a speed of 0.9 m/s.
- To find the boat's velocity relative to the bank, we need to subtract the current's velocity from the boat's velocity:
Velocity relative to the bank = Boat velocity - Current velocity
Velocity relative to the bank = 15 m/s - 0.9 m/s
Velocity relative to the bank = 14.1 m/s
b) The distance the boat is carried downstream:
- The boat's velocity relative to the bank is 14.1 m/s.
- The time it takes to cross the river is same as the time taken to be carried downstream.
- The distance carried downstream can be calculated using the formula:
Distance = Velocity x Time
Distance = 14.1 m/s x Time (Let's assume Time is T for now)
c) The time taken to cross:
- The boat is crossing a river with a width of 500 m.
- The boat's velocity relative to the bank is 14.1 m/s.
- The time taken to cross can be calculated by dividing the width of the river by the velocity:
Time taken to cross = Distance / Velocity
Time taken to cross = 500 m / 14.1 m/s
Time taken to cross = 35.46 seconds
d) The direction the boat must take to cross directly:
- To cross directly, the boat needs to take a direction that is opposite to the direction of the current.
- For this, the angle between the boat's velocity relative to the water and the river current should be 180 degrees.
- We need to find the angle between the boat's velocity relative to the water and the current.
- To find this angle, we can use the tangent trigonometric function:
Tangent(angle) = Current velocity / Boat velocity (relative to the bank)
Tangent(angle) = 0.9 m/s / 14.1 m/s
Angle = atan(0.9 / 14.1)
Angle ≈ 3.83 degrees
e) The time taken to cross directly, if it took the direction calculated in part d:
- Since the boat took the direction opposite to the current, it will not be carried downstream.
- The boat's velocity relative to the bank is still 14.1 m/s.
- The time taken to cross directly can be calculated using the formula:
Time taken to cross directly = Distance / Velocity
Time taken to cross directly = 500 m / 14.1 m/s
Time taken to cross directly ≈ 35.46 seconds
So, the solutions are:
a) The boat velocity relative to the bank is 14.1 m/s.
b) The boat is carried downstream for a distance of 14.1 m/s multiplied by the time taken to cross.
c) The time taken to cross the river is 35.46 seconds.
d) The boat must take a direction that is about 3.83 degrees opposite to the current to cross directly.
e) If the boat takes the direction calculated in part d, it will take approximately 35.46 seconds to cross directly.
To solve this problem, we can apply the concept of vector addition and relative velocity.
a) The boat velocity relative to the bank can be calculated by finding the vector sum of the boat's velocity in still water and the velocity of the current.
Boat velocity relative to the bank = Boat's velocity in still water + Velocity of the current
Given:
Boat's velocity in still water = 15 m/s
Velocity of the current = 0.9 m/s
Boat velocity relative to the bank = 15 m/s + 0.9 m/s = 15.9 m/s
b) The distance the boat is carried downstream can be calculated by multiplying the boat velocity relative to the bank by the time taken to cross the river.
Distance = Boat velocity relative to the bank x Time
Given:
Boat velocity relative to the bank = 15.9 m/s (from part a)
Time = Distance / Boat velocity relative to the bank
Time = 500 m / 15.9 m/s = 31.45 s
c) The time taken to cross the river is the same as the time calculated in part b.
d) The direction the boat must take, relative to the water if it is to cross directly is perpendicular to the current. This means the boat should be directed at a right angle to the current, towards the opposite bank.
e) The time taken to cross directly, if it took the direction calculated in part b, would be the same as the time calculated in part b, which is 31.45 seconds.
Therefore, the solutions to the problem are:
a) The boat velocity relative to the bank is 15.9 m/s.
b) The distance the boat is carried downstream is 500 meters.
c) The time taken to cross the river is 31.45 seconds.
d) The boat must take a direction perpendicular to the current, towards the opposite bank.
e) The time taken to cross directly, if it took the direction calculated in part b, is 31.45 seconds.