How to Calculate Relative Velocity Using Vector Addition

How do you calculate relative velocity using vector addition?

Billy-Bob and Jim set out in their pickup trucks from Tuscola, IL. Billy-Bob goes North at 60 mph; Jim goes East at 80. How fast are they moving relative to each other?

Calculation of Relative Velocity Using Vector Addition

Billy-Bob and Jim are moving relative to each other at a speed of 100 mph and at an angle of 36.87° from the east.

When calculating the relative velocity between two moving objects, we can use vector addition. In the case of Billy-Bob and Jim going in different directions, we need to consider their individual velocities and directions to find the resultant relative velocity.

Billy-Bob is moving north at 60 mph, represented as a vector with magnitude 60 mph and direction pointing north. Jim is moving east at 80 mph, represented as a vector with magnitude 80 mph and direction pointing east.

By using vector addition, we can find the magnitude of the resultant vector by taking the square root of the sum of the squares of the individual velocities:

Magnitude = √(60^2 + 80^2) = √(3600 + 6400) = √10000 = 100 mph

To determine the direction of the resultant vector, we can use trigonometry. The angle can be found by taking the inverse tangent of the ratio of the vertical component (north) to the horizontal component (east):

Angle = arctan(60/80) = arctan(0.75) ≈ 36.87°

Therefore, the relative velocity between Billy-Bob and Jim is 100 mph at an angle of 36.87° from the east. This calculation allows us to understand how fast they are moving relative to each other, taking into account both speed and direction.