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Vertical projectile motion - speed, time, height calculator

With this calculator, you can calculate the velocity of a body thrown vertically after a specified time and the initial velocity given to the body. You will also calculate the time elapsed from the moment of the throw to reach a certain speed, the time the body flies to the return to the place of the throw and the time after which the body reaches its maximum height. The calculator will also help you calculate the height at which the body is located after a certain time and will calculate the maximum height that the body will reach. The calculator also allows you to estimate the gravitational acceleration which can be useful in determining where the body was thrown.

Vertical projectile motion - speed, time, height

Vertical projectile motion - speed, time, height - information

Vertical projection - movement in the Earth's gravitational field with an initial velocity vertically upwards and the adopted approximations:
- resistance to movement is ignored,
- the throwing velocity is so low that the height achieved is much smaller than the ground radius (which allows the assumption that the gravitational field is homogeneous),

Vertical projection for flight stages:
1) at the beginning the body is at zero height (on the ground) with no movement
2) then, after giving the initial velocity, the body moves up with a uniformly delayed motion (due to the influence of gravity it flies up but slower and slower)
3) reaches its highest point of flight (maximum altitude) for a small fraction of a second
4) begins to fall with a uniformly accelerated motion (exactly as in free fall)
5) we end the examination of the vertical projection after falling to the ground (or another planet). Sometimes, however, problems are considered arising from the assumption that the body may fall further (e.g. because the body initially took off from a platform carried by balloons).

Velocity after time t from the throw up is given by the formula:
$$ v = v_0 - g · t $$
Formula for initial velocity is obtained after transforming the above:
$$ v_0 = v + g · t $$
Formula for throw time for a given velocity:
$$ t = \frac {v_0 - v}{g} $$
Knowing that the velocity of body at the highest point will be zero $v = 0 $, we can calculate the time of ascent that is the time after which the body reaches maximum height
$$ t_w = \frac{v_0}{g} $$
Flight time until it hits the baseline is twice as long as it reaches its maximum altitude: $$ t_s = 2 · t_w = \dfrac{v_0 + \sqrt{v_0^2 +2 g h_0}}{g} = \frac{2 · v_0}{g} $$
Height at which the body is located after a certain time since it was thrown up:
$$ h = v_0 · t - \frac {g · t^2}{2} $$
Maximum height reached by the body within the specified time since being thrown up:
$$ H_{max} = \frac {v_0^2}{2 · g} $$

where:
$ g $ - acceleration (e.g. $ earth = 9,81 \frac{m}{s^2}, lunar = 1,62 \frac{m}{s^2} $),
$ v $ – velocity after a specified time,
$ v_0 $– initial velocity given to the body,
$ t $- time,
$ t_w $- time of ascent (czas wznoszenia ciała do osiągnięcia wysokości maksymalnej),
$ t_s $- flight time until falling to the starting level,
$ h $- height at which the body is located after a certain time after being thrown up,
$ H_{max} $- maximum height that the body will reach from the moment it is thrown up,

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