A man who can swim at 5km/h in still water swims towards the east to cross arriver. If the river flows from north to south at the rate of 3km/h
a) Calculate:
i) The resultant speed
ii) The drift
b) If the width of the river is 30m, find the time taken, in seconds, for the man to cross the river
X = 5 km/h
Y = -3 km/h
i. Speed = Sqrt(X^2+Y^2) = sqrt(5^2+3^2)= 5.83 km/h = 1.62 m/s.
b. Tan A = Y/X = -3/5 = -0.60
A = -31o = 31o S. of E.
d = 30m/Cos(-31) = 35 m. = Distance to cross.
d = V*t = 35 m.
t = 35/V = 35/1.62 = 21.6 s.
a)
i) The resultant speed can be calculated using the Pythagorean theorem.
The speed of the man in still water is 5 km/h, and the speed of the river is 3 km/h. We can treat these speeds as vectors, with the speed of the man pointing east and the speed of the river pointing south.
Using the Pythagorean theorem, the resultant speed can be calculated as:
Resultant speed = sqrt((speed of the man)^2 + (speed of the river)^2)
= sqrt((5^2) + (3^2))
= sqrt(25 + 9)
= sqrt(34)
≈ 5.83 km/h (rounded to two decimal places)
ii) The drift can be calculated using trigonometry.
The drift is the component of the resultant speed that is perpendicular to the direction of the man's swimming. In this case, it is the component of the resultant speed that is in the north-south direction.
Using trigonometry, we can calculate the drift as:
Drift = (speed of the river) x (sin(angle between the man's direction and the river's direction))
The angle between the man's direction (east) and the river's direction (south) is 90 degrees.
So, the drift can be calculated as:
Drift = 3 km/h x sin(90 degrees)
= 3 km/h
b) To find the time taken for the man to cross the river, we can use the formula:
Time = (distance across the river) / (resultant speed)
The distance across the river is given as 30 meters and the resultant speed is calculated as 5.83 km/h.
Converting 5.83 km/h to m/s:
5.83 km/h = (5.83 x 1000) / (60 x 60) m/s
≈ 1.62 m/s (rounded to two decimal places)
Using the formula, the time taken can be calculated as:
Time = 30 m / 1.62 m/s
≈ 18.52 seconds (rounded to two decimal places)
To calculate the resultant speed of the man crossing the river, we need to use vector addition. The man's swimming speed in still water is 5 km/h towards the east, while the river flows from north to south at a rate of 3 km/h. Since these velocities act in different directions, we have to add them using vector addition.
a) i) Calculate the resultant speed:
To find the resultant speed, we can use the Pythagorean theorem. Since the man is swimming east and the river is flowing south, these velocities are perpendicular to each other. Therefore, we can calculate the resultant velocity by finding the hypotenuse of a right triangle formed by the man's swimming speed and the river's flow.
Using the Pythagorean theorem, the resultant speed is given by:
Resultant speed = sqrt((man's swimming speed)^2 + (river's flow speed)^2)
Plugging in the values:
Resultant speed = sqrt((5 km/h)^2 + (3 km/h)^2)
= sqrt(25 + 9)
= sqrt(34) km/h (rounded to two decimal places)
Therefore, the resultant speed of the man crossing the river is approximately 5.83 km/h.
a) ii) Calculate the drift:
The drift is the horizontal displacement caused by the river's flow while the person is swimming towards the east. To calculate the drift, we can use the formula:
Drift = river's flow speed * time
Since we are not given the time, we cannot calculate the drift at this point.
b) To find the time taken for the man to cross the river:
We can use the width of the river and the resultant speed to find the time taken to cross the river.
The speed at which the person is crossing the river can be given by calculating the horizontal component of the resultant speed. Since the swimmer is crossing east and the river is flowing south, the horizontal component of the resultant speed is equal to the man's swimming speed.
Crossing speed = man's swimming speed = 5 km/h
To find the time taken to cross the river, we can use the formula:
Time taken = Distance / Speed
Given that the width of the river is 30m (or 0.03 km) and the crossing speed is 5 km/h, plug in these values:
Time taken = 0.03 km / 5 km/h
= 0.006 hours
To convert hours to seconds, multiply by 60 minutes/hour and 60 seconds/minute:
Time taken = 0.006 hours * 60 minutes/hour * 60 seconds/minute
= 21.6 seconds
Therefore, it takes approximately 21.6 seconds for the man to cross the river.