On a windy day the wind speed is 4 m/s, in the south direction. A runner jogs west at 3 m/s. What is the velocity of the air relative to the runner to the nearest whole number? You may want to draw a small diagram.
To determine the velocity of the air relative to the runner, we need to consider the vector addition of the wind velocity and the runner's velocity.
Let's draw a diagram to represent the situation.
N
↑
|
W ←———R————→ E
|
↓
S
In this diagram, the north direction is upwards, the south direction is downwards, the west direction is towards the left, and the east direction is towards the right.
The wind is blowing in the south direction with a speed of 4 m/s, indicated by the vector "W". The runner is jogging in the west direction with a speed of 3 m/s, indicated by the vector "R".
To find the resultant velocity, we need to find the sum of the vector "W" and vector "R". Since both vectors are perpendicular to each other, we can use the Pythagorean theorem to calculate the magnitude of the resultant vector.
Using the Pythagorean theorem, we have:
Magnitude of the resultant vector = √(magnitude of vector W)^2 + (magnitude of vector R)^2
= √(4 m/s)^2 + (3 m/s)^2
= √16 m^2/s^2 + 9 m^2/s^2
= √25 m^2/s^2
= 5 m/s
The resultant vector has a magnitude of 5 m/s.
To determine the direction, we use the tangent of the angle between the resultant vector and the west direction.
Tangent of the angle = (magnitude of vector W) / (magnitude of vector R)
= 4 m/s / 3 m/s
= 1.333
Taking the inverse tangent (or arctangent) of 1.333, we find that the angle is approximately 53.13 degrees.
Therefore, the velocity of the air relative to the runner is approximately 5 m/s in a direction 53.13 degrees south of west.