A boat can travel at 3.1 m/s in still water. If the boat points its bow directly across a streamwhere the current is 1.0 m/s, a) what is the velocity of the boat relative to the shore?b) what will be the position of the boat, relative to its starting point, after 4.0 s?(Answer: 3.3 m/s [72o from shore], 12 m across & 4 m downstream)

a) The velocity of the boat relative to the shore can be found using the Pythagorean theorem, as the boat's velocity and the stream's velocity are at right angles to each other.

The boat's velocity (Vb) is 3.1 m/s and the stream's velocity (Vs) is 1.0 m/s.

The total velocity (Vt) can be calculated as follows:

Vt = sqrt(Vb^2 + Vs^2) Vt = sqrt((3.1 m/s)^2 + (1.0 m/s)^2) Vt = sqrt(9.61 m^2/s^2 + 1 m^2/s^2) Vt = sqrt(10.61 m^2/s^2) Vt = 3.3 m/s

The direction of the velocity can be found using trigonometry. The angle θ from the shore (opposite the current) can be found using the tangent function:

tan(θ) = Vs / Vb tan(θ) = 1.0 m/s / 3.1 m/s θ = arctan(0.32) θ = 72 degrees

So, the velocity of the boat relative to the shore is 3.3 m/s, 72 degrees from the shore.

b) The position of the boat relative to its starting point after 4.0 s can be found by multiplying the time by the components of the velocity.

The distance across the stream (Dx) is the boat's velocity times the time:

Dx = Vb * t Dx = 3.1 m/s * 4.0 s Dx = 12.4 m

The distance downstream (Dy) is the stream's velocity times the time:

Dy = Vs * t Dy = 1.0 m/s * 4.0 s Dy = 4.0 m

So, after 4.0 s, the boat is 12 m across and 4 m downstream from its starting point.