Suppose a river flows due south with a speed of 2.0 m/s. Given that speed of motorboat relative to water is 4.2 m/s due east.
(a) In what direction should the motorboat head to reach a point on the opposite bank directly east from your starting point?
(b) What is the velocity of the boat relative to the earth?
(c) If you travel a distance of 800 m in the water in total to cross the river, find the time is required to cross the river?
a. Vm - 2i = 4.2.
Vm = 4.2 + 2i = 4.65m/s[25.6o] N. of E. = Velocity and Heading.
b. Vm = 4.65m/s[25.6o].
c. d = Vm*t = 800.
t = 800/Vm = 800/4.65 = 172 s.
To solve this problem, we can break it down into different components and apply vector addition. Let's go step by step:
(a) To reach a point on the opposite bank directly east from the starting point, we need to take into account both the river flow and the velocity of the motorboat relative to the water.
1. First, draw a diagram representing the situation. Draw a line to represent the river flow (southward) with an arrow pointing down. Draw another line to represent the velocity of the motorboat (eastward) with an arrow pointing to the right.
2. To calculate the direction in which the motorboat should head, we need to find the resultant vector formed by the combination of the river flow vector and the motorboat velocity vector.
3. Use the Pythagorean theorem to find the magnitude of the resultant vector:
magnitude = √(velocity of motorboat^2 + velocity of river^2)
magnitude = √(4.2^2 + 2.0^2) = √(17.64 + 4) = √21.64 = 4.65 m/s
4. Use trigonometry to find the direction of the resultant vector (angle with respect to the east):
angle = arctan(velocity of river / velocity of motorboat)
angle = arctan(2.0 / 4.2) ≈ 26 degrees south of east
Therefore, to reach a point directly east, the motorboat should head approximately 26 degrees south of east.
(b) The velocity of the boat relative to the earth is the vector sum of the velocity of the motorboat relative to water and the velocity of the river.
1. Use vector addition to find the sum of the two velocity vectors. Since they are at right angles, we can use the Pythagorean theorem again:
magnitude = √(velocity of motorboat^2 + velocity of river^2)
magnitude = √(4.2^2 + 2.0^2) = √(17.64 + 4) = √21.64 = 4.65 m/s
2. The direction of the boat relative to the earth is the same as the angle found in part (a), which is 26 degrees south of east.
Therefore, the velocity of the boat relative to the earth is 4.65 m/s, approximately 26 degrees south of east.
(c) To find the time required to cross the river, we can use the formula: time = distance / velocity.
1. The distance traveled in the water is given as 800 m.
2. The velocity of the boat in the water is the velocity of the motorboat relative to water, which is given as 4.2 m/s.
3. Plug in the values into the formula:
time = 800 m / 4.2 m/s ≈ 190.48 seconds
Therefore, it would take approximately 190.48 seconds to cross the river.