Velocity in Two Dimensions

The actual range of a projectile is a bit complicated. It is calculated as the initial velocity squared multiplied by the sine of 2 times the angle all divided by the force of gravity. The assumption is that the range is small compared to the circumference of the earth. If the range is large, the earth will curve away and the acceleration of gravity changes direction. Technically, if the speed is initially great enough, the projectile will go into orbit. This is called the exit velocity.

You can calculate the time of flight of a projectile that is launched and lands on a flat horizontal surface, which works for distances that make the earth’s curvature negligible. In such cases, the starting point with respect to y is zero and the ending point with respect to y is zero. With the displacement being zero, the time of flight is going to be calculated as 2 times the initial velocity times the sine of the initial angle divided by the force of gravity.

VELOCITY IN TWO DIMENSIONS

Velocity in two dimensions can be described when someone tries to cross a stream in a boat but gets caught up in the current. The boat is then moving in a direction in which it is not pointed. The same would have to be said of an airplane stuck in a crosswind. There is a straight direction with respect to the air but not relative to the ground. In such cases, there will be two velocity vectors that need to be added to get the actual velocity.

Remember that velocity is a vector so that the rules of vector velocity addition and subtraction still apply. Remember, too, that velocities in the x and y direction can be thought of as being separate so that the velocity in the x direction is the total velocity multiplied by the cosine of theta, while the velocity in the y direction is thought of as the total velocity multiplied by the sine of theta. Figure 13 shows the velocity equations:

Figure 13.

Graphing of velocities is identical to displacement in two directions, with vectors on the x and y axis showing the velocities in both directions perpendicular to one another adding to make the relative velocity.

Let’s do one problem. A boat is traveling at 0.75 meters per second in a current that is going across it to the right at 1.2 meters per second. What is the relative velocity of the boat relative to the shore? Start by getting the total velocity, knowing that the Vx velocity is 1.2 meters per second and the Vy velocity is 0.75 meters per second. This is shown in figure 14:

Figure 14.

The total velocity will be the square root of the sum of the squares of Vx and Vy or, using a calculator, you get 1.42 meters per second. The angle, according to the calculations, is 32 degrees. If you know the total width of the river, you can use the angle and the width (which will be the y direction in the figure) in order to get the x direction or the point on the river that the boat will reach at the end of the trip. This will be the displacement in the x direction after how many seconds it takes to cross the span.

For example, if the river is 100 meters in width, it will take 0.75 meters per second or 133.3 seconds to cross. In that same 133 seconds, it would traverse 1.2 meters per second multiplied by 133.3 seconds or about 160 meters downstream. This is simple math because the river crossing is the y-axis and the distance downstream is the x-axis.

When adding velocities, the relative velocities will be relative to a certain reference frame. This concept of relative velocities is one aspect of relativity, which is the study of how various observers relative to one another will measure the same phenomenon. On an airplane, the passengers have a relative velocity of zero (as they do not perceive movement relative to one another); there will also be velocity relative to the wind and velocity relative to the ground. These are examples of classical relativity, which apply when speeds are less than about 1 percent of the speed of light, which is less than 3000 kilometers per second. Modern relativity (Einstein’s relativity) applies at higher speeds, which will be discussed later.