Ships A and B leave port together. For the next two hours, ship A travels at 30.0 in a direction 75.0 west of north while the ship B travels 25.0 east of north at 40.0 . What is the distance between the two ships two hours after they depart? What is the s

(-6.51,76.8)

To find the distance between the two ships after two hours, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we have two sides of the triangle: the northward distance traveled by ship A and the northward distance traveled by ship B. Let's calculate these distances:

Distance traveled by ship A in 2 hours:
Speed of ship A = 30.0 mph
Time = 2 hours
Distance = Speed × Time = 30.0 mph × 2 hours = 60.0 miles north

Distance traveled by ship B in 2 hours:
Speed of ship B = 40.0 mph
Time = 2 hours
Distance = Speed × Time = 40.0 mph × 2 hours = 80.0 miles north

Now, we can use the Pythagorean theorem to find the distance between the two ships:

Distance between the two ships = sqrt((Distance A)^2 + (Distance B)^2)
= sqrt((60.0 miles)^2 + (80.0 miles)^2)
= sqrt(3600.0 miles^2 + 6400.0 miles^2)
= sqrt(10000.0 miles^2)
= 100.0 miles

So, the distance between the two ships two hours after they depart is 100.0 miles.

Please note that the question asks for the distance and not the displacement between the ships. The displacement may be different depending on the coordinate system used.

To find the distance between the two ships after two hours, we can use the concept of vector addition. The first step is to find the displacement vectors for each ship.

Ship A travels at a speed of 30.0 m/s in a direction 75.0° west of north. To find the displacement, we can use trigonometry.

The vertical component of the displacement vector for ship A is calculated by:
Vertical displacement = 30.0 m/s * sin(75.0°)

The horizontal component of the displacement vector for ship A is calculated by:
Horizontal displacement = 30.0 m/s * cos(75.0°)

Similarly, ship B travels at a speed of 40.0 m/s in a direction 25.0° east of north.

The vertical component of the displacement vector for ship B is calculated by:
Vertical displacement = 40.0 m/s * sin(25.0°)

The horizontal component of the displacement vector for ship B is calculated by:
Horizontal displacement = 40.0 m/s * cos(25.0°)

Now, to find the total displacement between the two ships, we can add the corresponding components of the displacement vectors:

Vertical displacement = Vertical displacement of ship A - Vertical displacement of ship B
Horizontal displacement = Horizontal displacement of ship A + Horizontal displacement of ship B

Finally, we can use the Pythagorean theorem to find the distance between the two ships:

Distance = sqrt((Vertical displacement)^2 + (Horizontal displacement)^2)

After 1 hour, A is at (30*cos 165,30*sin 165) = (-28.98,7.76)

After 1 hour, B is at (40*cos 50,40*sin 50) = (25.71,30.64)

Complete the math for your answer.