A ship is travelling with a velocity (30i- 42j), and a trolley is being pushed relative to the ship with the velocity (-5i- 6j). A fly is crawling on the trolley with relative velocity ( -i- 1/2j - k).
a) calculate the velocity and the speed of the relative to the Sea. all velocities in this question are given in km/h.
b) a bird is flying overhead with a velocity vector 22i + 10j - 6k
what is the angle between the velocity vectors of the bird and the fly?
**every letter next to a number should have an arrow on top showing it is a vector!
To calculate the velocity of the trolley relative to the sea, we need to add the velocity of the ship to the velocity of the trolley relative to the ship.
a) Velocity of the trolley relative to the sea:
V_trolley_sea = V_ship + V_trolley_ship
Given:
V_ship = 30i - 42j km/h
V_trolley_ship = -5i - 6j km/h
Adding the two vectors:
V_trolley_sea = (30i - 42j) + (-5i - 6j)
= (30 - 5)i + (-42 - 6)j
= 25i - 48j km/h
So, the velocity of the trolley relative to the sea is 25i - 48j km/h.
To calculate the speed of the trolley relative to the sea, we need to find the magnitude of the velocity vector.
b) Speed of the trolley relative to the sea:
Speed_trolley_sea = |V_trolley_sea|
Using the Pythagorean theorem:
Speed_trolley_sea = sqrt((25)^2 + (-48)^2)
= sqrt(625 + 2304)
= sqrt(2929)
≈ 54.11 km/h
So, the speed of the trolley relative to the sea is approximately 54.11 km/h.
To find the angle between the velocity vectors of the bird and the fly, we can use the dot product.
Given:
V_bird = 22i + 10j - 6k km/h
V_fly = -i - (1/2)j - k km/h
The dot product formula for two vectors A and B is:
A · B = A_x * B_x + A_y * B_y + A_z * B_z
Calculating the dot product:
V_bird · V_fly = (22i + 10j - 6k) · (-i - (1/2)j - k)
= (22 * -1) + (10 * (-1/2)) + (-6 * -1)
= -22 - 5 - 6
= -33
The magnitude of a vector A is given by:
|A| = sqrt(A_x^2 + A_y^2 + A_z^2)
Calculating the magnitudes:
|V_bird| = sqrt((22)^2 + (10)^2 + (-6)^2)
= sqrt(484 + 100 + 36)
= sqrt(620)
≈ 24.9
|V_fly| = sqrt((-1)^2 + (-1/2)^2 + (-1)^2)
= sqrt(1 + 1/4 + 1)
= sqrt(9/4)
= 3/2
The angle between two vectors A and B is given by:
θ = arccos((A · B) / (|A| * |B|))
Calculating the angle:
θ = arccos(-33 / (24.9 * 3/2))
≈ arccos(-1.32)
Note: The arccos function returns an angle in radians. To convert it to degrees, you can multiply by (180/π).
So, the angle between the velocity vectors of the bird and the fly is approximately -75.3 degrees.