Velocity is speed with a direction

In physics, we often associate a direction with physical quantities. For example, it's nice to know the direction in which an object is moving or in what direction a force is applied to an object. To do this, we employ the concept of vectors. I strongly suggest that if you have not, you take a look at the vectors section before working through this one. It's absolutely crucial in physics to have a good working understanding of vectors.

Velocity is the vector version of speed. Velocity is speed with a specified direction.

A velocity vector is an arrow that points in the direction of motion and has a length proportional to the speed. We call the speed the magnitude (size) of the velocity vector.

The only two things of importance about vectors is length (magnitude) and direction. Velocity vectors can be moved around in a plane or in 3-D space at will for the convenience of making calculation easier, so long as length and direction aren't affected.

A word about vector notation

Often in textbooks, vectors are denoted by symbols either in bold-face or with an arrow overhead, so the velocity vector would be v or v. I am going to choose not to use any special notation for vectors throughout these notes. It will not only make the writing simpler, but it also forces the reader to be mindful about what is going on ... what's a vector and what is not ? ("not vectors" are "scalars"). That kind of mindfulness—about what is physically happening, is a key to success in physics.

Why velocity vectors?

The beauty of the velocity vector is that it lets us take velocity in any arbitrary direction and split it (or resolve it) into components that lie along (parallel to) the axes of our coordinate system. Let the vector v, below, be a velocity vector:

Now let's resolve vector v into its components along each of the coordinate (x & y) axes. We do this in two steps:

1. Move the vector to the origin

2. Find the components, vx and vy along the x and y axes.

Here's how it looks on the graph:

Notice that the vector sum of vx and vy, added in either order because vector addition is commutative, is just our velocity vector v.

This method of decomposing vectors into two new vectors aligned with our coordinate axes will be invaluable as you continue to study physics, so learn it well. Re-read the vectors notes if you need to.

Vectors vs. scalars

Velocity is a vector and speed is a scalar. Speed is the length of the velocity vector, calculated using the Pythagorean theorem:

A vector has length and direction, a scalar is just a number.

Speed is the length (also called "magnitude" or "modulus") of the velocity vector.

Examples: Airplane velocity vectors

Consider an airplane that can travel at a speed of 150 Km/h in still air. That would give it a velocity vector of 150 Km/h in length pointed in its direction of travel. Now add in a 30 Km/h headwind. The situation is shown below. The net forward velocity of the plane is the sum of 150 Km/h and -30 Km/h, or 120 Km/h in the direction of travel.

Now let's turn that headwind into a tailwind, or pushing wind. The vector diagram is shown below. This time the 30 Km/h wind vector adds to the velocity of the plane, for a total velocity of 180 Km/h.

Now let's get a little fancier. How about a cross-wind, or a wind that comes from 90˚ from the direction of travel – from the right or left.

To add these two vectors, just use the Pythagorean theorem. You can see that the direction of travel will be slightly to the right of where the plane is steering, and it's velocity will be 6 Km/h faster than without the crosswind.

Finally, let's let the wind hit the plane at an odd angle, so that if we want the new velocity vector, we have to solve a law of cosines problem. If you don't know the LOC, don't worry about it right now.

This is basically how we navigate planes without more sophisticated equipment like radio and radar.

Practice problems

  1. On a windless day, a boat can travel at a maximum forward speed of 10 knots. It travels across a lake on a day with a 2-knot cross wind (perpendicular to the intended direction of travel) from the right. Calculate the actual speed of the boat (in knots) and its deviation (in degrees) from its intended course.
  1. How fast can an airplane, with a maximum speed of 160 Km/h in still air, travel 250 miles if it has a steady headwind of 15 Km/h ?
  1. A projectile is fired upward at an angle of 45˚ with an initial velocity of 30 m/s. Calculate the magnitude of the component velocity vectors along the ground and straight up (x and y, respectively).
  1. Two cars begin a trip together, one traveling at 65 mi./h, and the other at 75 mi./h on the same freeway. After how many miles will the slower car be 1 hour behind the faster?

More vector resolution: Trigonometry

Very often we need to resolve a vector of known magnitude that forms a known angle with the x- or y-axes of the Cartesian coordinate system. For that we use simple right-angle trigonometry.

The figure on the right shows the trigonometric relationships that allow us to find the x and y components of any vector. This idea is completely generalizable to 3-D (or higher!) coordinates.

If the components of such a vector are known but the vector angle is not, the angle can be found using the tangent function. Here's an example, a velocity vector with a magnitude of 25 m/s (speed = 25 m/s), oriented at 25˚  to the x-axis.

Now we'll decompose this vector into two component vectors along the x- and y-axes.

Using basic right triangle trigonometry, we can calculate vx:

Similarly, we calculate vy:

So if you were standing on the y-axis, this object would be moving away from you at 23.3 m/s, and if you were on the x-axis, it would be moving away from you at 11.3 m/s. "Moving away" is defined here as getting farther measured perpendicular to the y, x axes.

Practice problems

  1. A jet flies at 600 Km/h in still air at maximum power. The same jet using maximum power to fly due west (270˚ on the compass), flies into a 100 Km/h northwest wind that makes a 23˚ angle with the intended path (293˚ on the compass. Find the actual velocity of the plane.
  1. If a swimmer can swim at 0.5 m/s in still water, how long will it take her to swim across a 50 m-wide river flowing at 0.5 m/s? Assuming she swims perpendicular tot he flow of the river, how far downstream will she be when she reaches the shore?
  1. A projectile is launched with an initial velocity of 29 m/s and at an angle of elevation with respect to the ground of 39˚. Calculate the component of the projectile velocity parallel to the ground.
  1. A plane flies with a velocity of 53 m/s east (90˚ on the compass), encountering a crosswind from 10˚ (that's 10˚ to the 'right' of north, which is 0˚). Calculate the magnitude and direction (relative to due north, 0˚) of the resultant velocity at which it travels.

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