Time goes by at a constant speed

Constant speed movements

Multi-dimensional movements with constant speed¶

The laws for one-dimensional movements presented in the last section can also be transferred to two-dimensional movements without great effort. A basic principle here is that every two-dimensional movement process is converted into a - and a -Component can be split. The alignment of the coordinate system can in turn be freely selected once, but is then binding for the rest of the calculation.

A second basic principle is that individual movement processes, even if they take place along different spatial directions, can each be viewed separately from one another.

Addition of partial speeds

If two movements run in a straight line along a common line, a simple addition of the two amounts of speed is sufficient and to get the resulting speed.

Examples:

  • A person moves at one speed on a treadmill against the treadmill speed . If both speeds are the same, the person stays in the same place - the resulting speed is zero.

    If the two speeds are different, the person moves in the direction of the higher speed. [1]

  • If the direction of movement of the person corresponds to the direction of the treadmill speed, the amounts of both speeds add up. The speed the person (relative to the ground) is thus the same .

The addition of the occurring speeds is also possible if they are at any angle to each other. To do this, one draws the two speeds and as arrows, the directions of which coincide with those of the two speeds and the lengths of which depict the amounts of both speeds. According to the rules of vector addition, the direction and the amount of the resulting speed can be determined from the two speed arrows determine graphically.

Amount and direction of the resulting speed can also be determined mathematically. For a two-dimensional movement (in one plane) the following applies:

The resulting speed thus corresponds to a component-wise addition of the two speed vectors and . For the amount of the resulting speed applies:

From the relationship of the - to Component can be used to determine the angle of the resulting speed:

Example:

  • A boat crosses at one speed perpendicular to a river that flows at a speed flows. If you create a coordinate system in such a way that the -Axis in the direction of the speed of the boat and the -Axis points in the direction of the flow of the river, it follows for the resulting velocity :

    The amount of the resulting speed is here:

    The angle opposite the -Axis (direction of the river) is:

    The boat is thus drifting around an angle of around from.

The distances covered in the respective directions and can in turn be calculated component-wise using the formula to calculate.

A further generalization to three-dimensional movement processes is made according to the same principles by adding a -Component and therefore calculates with three- instead of two-dimensional vectors.


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