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:
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