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VERTICAL THROW: EXPLANATION, TYPES & FORMULAS

14 de septiembre, 2022Por Cultural Maya Tips

Only one point separates the second in the world rankings from winning the set
and thus winning the match. Does he keep a calm head in this situation? He
swings. He throws the ball and hits it. And? The crowd cheers. An ace!

This situation is a possible comment on a tennis match. The serve is a central
component with which the players can get a direct point in the match.

The ideal throw for a serve is a vertical throw. In this article you will find
out what this means and how you calculate it.


VERTICAL THROW – PHYSICS

Tennis players swing their arms up with the tennis ball and release the ball
vertical with a certain initial speed at a height or go. First, the tennis ball
continues to gain altitude. But at a certain point, it no longer flies upwards,
but sinks again until it finally lands on the ground. This describes the
throwing movement of a vertical throw up with initial height. But there are also
other types of vertical throw.


VERTICAL THROW – DIVISION OF MOVEMENT

The tennis ball can be thrown vertically upwards when it serves, so that a
vertical throw is carried out. Some players prepare to serve by dribbling the
ball in front of them. Again, that would be a vertical throw, however downward.
The following classification can be made for a vertical throw:

The throwing movement of the vertical throw is seen as a superimposition of two
partial movements. In the following you will find out why this is the case and
how the throwing movement can be described and calculated.


VERTICAL THROW – BASIC KNOWLEDGE

The course of a movement over time and the resulting speeds and accelerations
are decisive for the category in which a movement can be divided. If a body is
moving at a constant speed (i.e. it is neither getting faster nor slower), then
it is performing a uniform movement. This is also shown in the left part of
Figure 3.

If the speed of a body is not constant during a movement (i.e. it becomes faster
or slower), then the movement falls into the category of non-uniform movement.
You can choose between movements with constant acceleration (uniformly
accelerated movement) and non-uniformly accelerated movements be
distinguishedwhere the acceleration is not constant.

The throwing movement of the vertical throw is a superimposition of two partial
movements:

 * a uniform movement in positive y-direction (up or down)
 * and a uniformly accelerated movement (free fall) in the negative y-direction
   (downwards)

This superimposition is possible due to the so-called superposition principle
or. principle of independence. This states that partial movements can overlap to
form an overall movement if they do not influence each other.



In the table you will find a brief repetition of the most important quantities
and properties of these two types of movement.

Depending on the orientation of the coordinate system in free fall the formulas
are adjusted. This is also relevant for the vertical throw, as will be shown
below.

You can also read all information about the types of movement in the articles on
«uniform motion» and «uniformly accelerated motion» or «free fall».


VERTICAL THROW – INFLUENCE OF WEIGHT

If a ball or other body is thrown vertically upwards, the height will increase
up to a certain point. The ball then reverses its flight direction and loses
altitude until it hits the ground. The reason for this is the force of gravity
or the force of gravity, which causes an attraction between massive objects and
the earth. According to Newton’s basic law, every body with the so-called
gravitational acceleration accelerated towards the center of the earth.

This gravitational acceleration corresponds approximately to the value of an
acceleration of . Since this attraction can never be switched off, a ball thrown
vertically upwards (as in Figure 4) can never continue to fly upwards
indefinitely and will therefore sooner or later sink to the ground again.
Therefore, a uniformly accelerated movement of the free fall are taken into
account.

In order to delve deeper into the topic, please read the articles on
«gravitational force» and «free fall»!


VERTICAL THROW UP – CALCULATION AND FORMULAS

In the classification you have already seen that the starting situation differs
for the vertical throw up and the vertical throw down. First to the vertical
throw up.


VERTICAL THROW UP – SPEEDS AND ACCELERATION

In order to be able to better understand the speeds of a vertical throw upwards,
the example of a tennis serve is used. Since only the initial height differs in
the two vertical throws upwards, the fall will occur with initial height
considered and later only adjusted the formula.

The speed is a directional variable, which is why it is shown with an arrow.
Since only the amount is usually relevant for calculations, the formulas are
shown without this arrow.

Before the movement starts, the ball is accelerated in the player’s hand and
then leaves the hand with a certain initial speed in the y-direction. The
starting speed of the uniform movement in the positive y-direction is therefore
declared as at the time.



For a better understanding, the index GB = uniform movement is used here.

If only the uniform motion would be considered, this speed would not change.



The overall speed can therefore be calculated at any point in time by combining
both partial formulas into one overall formula.

The speed of the ball or the body can be determined with it. The acceleration
should also be mentioned in this context. Since acceleration only exists in free
fall, the total acceleration can be summarized as follows:

acceleration when thrown vertically upwards depending on the acceleration due to
gravity:



With

Different speeds prevail at different points in time and thus positions of the
ball.


VERTICAL THROW UP – POSITION

In addition, there is the second partial movement of the free fall. Considered
individually, the position can be calculated as follows:

In order for the position to be calculated, the two partial movements must be
reassembled, resulting in the following:

the position when throwing vertically upwards with initial height depending on
the partial movements of the uniform movement and the free fall is calculated as
follows:



With

It doesn’t matter whether the ball is moving up or is already on the way down,
this formula can be used to determine the position of the body at any time.

It can also be a vertical throw without initial height is discarded, then . This
is accounted for in the formula by using the initial state. The following
applies:

To determine the maximum throw height, simply put the climb time into the
formula for calculating the y-position.

Maximum throw height when throwing vertically upwards without Initial height:



With

How long does it take for the ball to hit the ground again? For a vertical throw
with no initial height, this can be calculated by doubling the climb time. The
ball takes as long to rise as it takes to fall.

Derivation and formula Litter duration:



With

Any size of this throwing movement can be described and calculated using the
formulas and laws in the section on the vertical throw upwards. The
velocity-time laws as well as the acceleration-time laws and the location-time
laws can also be based on motion diagrams be visualized.


VERTICAL THROW UP – MOVEMENT DIAGRAMS

The following movement diagrams show how the quantities behave in the vertical
throw as a function of time. The diagrams for a vertical throw upwards with or
without an initial height differ only in the location-time diagram. The
speed-time diagram and the acceleration-time diagram are schematic almost
identical.

Vertical throw up with initial height

Figure 12: Movement diagrams of vertical throw upwards with initial height

Vertical throw up with no initial height

Figure 13: Movement diagrams of vertical throw upwards without initial height

As mentioned earlier, tennis players sometimes dribble before serving. The
movement is completely different, but it is also a vertical throw. How can the
sizes be determined in this case?




VERTICAL THROW DOWN – CALCULATION AND FORMULAS

For a vertical throw down, the same overlays apply as for a vertical throw up.
However, there is no up and down movement here, only a downward movement. How do
the partial velocities and accelerations behave in this case?


VERTICAL THROW DOWN – SPEEDS AND ACCELERATION

For a better understanding, the throwing movement of tennis players is shown and
all relevant sizes are drawn in the sketch. The choice of the coordinate system
must be taken into account. There is an option to place the origin of the
coordinate system on the ground and choose a y-axis up. The second possibility
would be to place the origin of coordinates at a certain height to start the
movement. Accordingly, you would have to put the y-axis down.

The derivation of the formulas for the axis orientation upwards is shown below.
You will find the formulas for orientation below in the definition boxes.

As with the vertical throw upwards, the ball is initially accelerated in the
hand of the tennis player and then leaves the hand with the initial speed, which
is directed downwards in the y-direction. This again corresponds to the speed of
uniform motion, only with a negative sign. But here, too, the gravitational
acceleration acts on the body, which is why there is a second partial speed in
the same direction. The following therefore applies to the axis orientation
upwards:



For a better understanding, the formulas are marked analogously to the vertical
throw upwards. FF = free fall,

GB = uniform motion.

The individual velocities must be recombined for a complete description of the
vertical throw down. The table shows the speeds at different times and periods.

In this context, the acceleration can also be determined again. The
gravitational acceleration also applies here. Therefore, only the axis
orientation has to be considered in the formula.

acceleration when throwing vertically downwards (Axis orientation upwards):



acceleration at the…


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