A ship is heading north at 15km/h in a tide moving at 6km/h westward. Determine the magnitude and direction of the resultant velocity of the ship.
with a little trig:
direction: arctan(15/6) W of N
magnitude: sqrt(15^2+6^2)
To determine the magnitude and direction of the resultant velocity of the ship, we can use the concept of vector addition.
Let's represent the northward velocity of the ship as vector A and the westward velocity of the tide as vector B.
Vector A has a magnitude of 15 km/h towards the north direction.
Vector B has a magnitude of 6 km/h towards the west direction.
To find the resultant velocity, we need to add these two vectors.
Step 1: Draw vector A in the north direction.
Step 2: Draw vector B in the west direction.
Step 3: Use a ruler or protractor to measure the length of vector A and vector B.
Step 4: Place the tail of vector B at the head of vector A, so that the arrowhead points from the tail of vector A to the head of vector B.
Step 5: Draw the resultant vector starting from the tail of vector A and ending at the head of vector B.
Step 6: Measure the magnitude and direction of the resultant vector.
Calculating the magnitude:
Applying the Pythagorean theorem, we can calculate the magnitude of the resultant velocity (R) with the formula:
R^2 = A^2 + B^2
where A and B are the magnitudes of vector A and B, respectively.
R^2 = (15 km/h)^2 + (6 km/h)^2
R^2 = 225 km^2/h^2 + 36 km^2/h^2
R^2 = 261 km^2/h^2
R = √(261 km^2/h^2)
R ≈ 16.16 km/h (rounded to two decimal places)
Calculating the direction:
Applying the trigonometric function of tangent, we can calculate the direction of the resultant velocity (θ) with the formula:
θ = arctan(B/A)
θ = arctan(6 km/h / 15 km/h)
θ = arctan(0.4)
Using a calculator or trigonometric tables, we find that:
θ ≈ 21.8° (rounded to one decimal place)
So, the magnitude of the resultant velocity is approximately 16.16 km/h and its direction is approximately 21.8° north of west.
To determine the magnitude and direction of the resultant velocity of the ship, we can use vector addition.
Step 1: Draw a diagram to represent the problem. Draw a northward arrow to represent the ship's velocity of 15 km/h and a westward arrow to represent the tide's velocity of 6 km/h.
Step 2: To add the two vectors, place the tail of the second vector (tide's velocity) at the head of the first vector (ship's velocity). In this case, it means placing the tail of the westward arrow at the head of the northward arrow.
Step 3: Draw the resultant vector, which is the vector that starts at the tail of the first vector (ship's velocity) and ends at the head of the second vector (tide's velocity). The length of this resultant vector represents the magnitude of the resultant velocity, while the direction represents the direction of the resultant velocity.
Step 4: To find the magnitude of the resultant velocity, use the Pythagorean theorem. The magnitude is the square root of the sum of the squares of the individual velocities. In this case, it would be the square root of (15^2 + 6^2).
Step 5: Calculate the magnitude of the resultant velocity: magnitude = sqrt(15^2 + 6^2) = sqrt(225 + 36) = sqrt(261) ≈ 16.12 km/h.
Step 6: To determine the direction of the resultant velocity, use trigonometry. The angle of the resultant vector can be found using tangent: angle = arctan(opposite/adjacent) = arctan(6/15) ≈ 22.62 degrees.
Therefore, the magnitude of the resultant velocity is approximately 16.12 km/h, and the direction is approximately 22.62 degrees west of north.