Calculating velocity at an angle is a trigonometry problem.
Velocity is a vector, which means it has both magnitude or length and direction. In math and physics, however, we often need to break the velocity vector down into its components — how fast an object is traveling with respect to the x-axis and how fast with respect to the y. Thanks to trigonometry, this kind of problem is not as difficult as you might think. If you have the angle between the velocity and another vector, you can use this in your calculations.
1. Measure the magnitude of the velocity vector and the angle between the velocity vector and the vector or axis of interest. Alternatively, if you’re working a math quiz question, you’ll probably be given this information.
2. Take the cosine of the angle between the velocity vector and the vector or axis of interest using the “cos” button on your calculator.
3. Multiply the value from the last step by the magnitude of the velocity vector. This operation gives the magnitude of the velocity in the direction of the vector of interest.
Example: Imagine a rocket traveling at 200 mph at an angle of 70 degrees above the horizontal. The rocket is cruising towards the summit of a hill whose surface is inclined at 45 degrees above the horizontal. How fast is the rocket traveling with respect to the hill’s surface (i.e. in the direction of the hill’s summit)?
Answer: The angle between the rocket’s velocity vector and the slope of the hill is 70 degrees – 45 degrees = 25 degrees. The cosine of 25 degrees is 0.906, therefore 0.906 x 200 mph = 181.26 mph.