The vector of speed characterizes the movement of a body, showing the direction and speed of movement in space. Speed as function is the first coordinate, derivative of the equation. Derivative of speed will give acceleration.
Instruction
1. In the itself set vector gives nothing in respect of the mathematical description of the movement therefore it is considered in projections to coordinate axes. It can be one coordinate axis (beam), two (plane) or three (space). To find projections, it is necessary to lower perpendiculars from the ends of a vector on an axis.
2. The projection represents kind of vector "shadow". If the body moves perpendicular to the considered axis, the projection will degenerate in a point and will have zero value. At the movement parallel to a coordinate axis the projection coincides with the vector module. And when the body moves so that its vector is sent to speed under some corner φ to axis x, the projection to axis x will be a piece: V (x) = V· cos(φ), where V – the speed vector module. The projection is positive when the direction of a vector of speed coincides with the positive direction of a coordinate axis, and is negative in the return case.
3. Let the movement of a point be set by the coordinate equations: x=x(t), y=y(t), z=z(t). Then functions of the speed projected on three axes will have an appearance, respectively, to V (x) =dx/dt=x' (t), V (y) =dy/dt=y' (t), V (z) =dz/dt=z' (t), that is for finding of speed it is necessary to take derivatives. The vector of speed will be expressed by the equation of V=V (x) • i+V(y) • j+V(z) • k where i, j, k are single vectors of coordinate axes x, y, z. The module of speed can be calculated on a formula V= √ (V (x) ^2+V(y) ^2+V(z) ^2).
4. Through the directing cosines of a vector of speed and single pieces of coordinate axes it is possible to set the direction to a vector, having rejected its module. For a point which moves to the planes there are enough two coordinates, x and y. If the body makes the movement on a circle, the direction of a vector of speed continuously changes, and the module can both remain to constants, and to change in time.