A boat heads directly across a river 39 m wide at 3.9 m/s. The current is flowing downstream at 2.4 m/s.

(a) What is the resultant velocity of the boat?
m/s (magnitude)
° (direction)
(b) How much time does it take the boat to cross the river?
s
(c) How far downstream is the boat when it reaches the other side?
m

To solve this problem, we can use vector addition and some basic trigonometry.

(a) The resultant velocity of the boat can be found by adding the velocity of the boat relative to the ground and the velocity of the current. Since the boat is heading directly across the river, the angle between the boat's velocity and the river's current is 90 degrees. Therefore, we can use the Pythagorean theorem to find the magnitude of the resultant velocity.

The magnitude of the resultant velocity (V) can be calculated using the formula:

V = √(Vb^2 + Vc^2)

where Vb is the velocity of the boat and Vc is the velocity of the current.

Substituting the values, Vb = 3.9 m/s and Vc = 2.4 m/s, we have:

V = √(3.9^2 + 2.4^2)
V ≈ √(15.21 + 5.76)
V ≈ √21.97
V ≈ 4.69 m/s

So, the magnitude of the resultant velocity of the boat is approximately 4.69 m/s.

To find the direction of the resultant velocity, we can use trigonometry. Since the boat is heading straight across the river, the angle between the resultant velocity and the side of the river is the same as the angle between the boat's velocity and the side of the river. This angle can be calculated as:

θ = tan^(-1)(Vc / Vb)

Substituting the values, we have:

θ = tan^(-1)(2.4 / 3.9)
θ ≈ tan^(-1)(0.6154)
θ ≈ 31.9°

Therefore, the direction of the resultant velocity is approximately 31.9°.

(b) To find the time it takes for the boat to cross the river, we can use the formula:

t = d / Vb

where t is the time, d is the width of the river, and Vb is the velocity of the boat.

Substituting the values, we have:

t = 39 m / 3.9 m/s
t = 10 s

So, it takes 10 seconds for the boat to cross the river.

(c) To find how far downstream the boat is when it reaches the other side, we can use the formula:

d_downstream = Vc * t

where d_downstream is the distance downstream, Vc is the velocity of the current, and t is the time.

Substituting the values, we have:

d_downstream = 2.4 m/s * 10 s
d_downstream = 24 m

Therefore, the boat is 24 meters downstream when it reaches the other side.