First, a bit of theory. We can distinguish couple of classic examples of physical body motion in gravitational field:
- upward projection and free fall
- horizontal projection
- angular projection
Below is description of each motion:
Ad 1. Upward projection and free fall
Depending on initial conditions (position and velocity), we can distinguish the following combination of those 2 motions:- upward motion and then free fall
- only free fall

Fig 1.
- Ad a. Ball is moving upward and then does free fall, when its initial velocity:
- Ad b. Ball does only free fall, when its initial velocity is →0 and initial position is above the ground. Parameters that describe the ball's motion are:
→V0=[0,Vy], where Vy>0.
We can define key parameters that describe ball movement:t0 - time when movement starts (here t0=0),
tmax - time, when ball reaches highest height above the Earth's surface,
tend - time, when ball falls to the ground (moment of contact with Earth's surface),
Ek - kinetic energy of ball,
Ep - potential energy of ball,
V0 - initial speed of ball (V0=|→V0|),
Vend - speed of ball, when it contacts the ground,
h - height of ball above the ground,
h0 - initial height of ball laying on the ground. It equals to its radius r. If r≪hmax, then h0≈0,
hmax - max height of ball above the ground,
g - acceleration of gravity (9.8 m/s2). In our simulation we omit force due to air resistance. We can use energy conservation law, where only factors are potential and kinetic energy. Therefore: Ep(tmax)=Ek(t0)=Ek(tend) mghmax=mV202=mV2end2 hmax=V202g=V2end2g We can come to conclusion: V0=Vend Let's calculate tmax, tend now. To do this, we need to use formula for h: h=h0+V0t−gt22 tmax corresponds to hmax, therefore: hmax=h0+V0tmax−gt2max2 Now, equation (1) needs to be put into formula (4). Finally tmax is: tmax=V0g Condition for tend looks like this: h0=h0+V0⋅tend−gt2end2 Therfore after simple calculations, tend is: tend=2V0g After comparison of tmax with tend, we can conclude that motion of ball, when it is first moving upward and then falls to the ground is completely symmetrical.
t0 - time when free fall motion starts,
tend - time, when ball falls to the ground (moment of contact with Earth's surface),
Vend - speed of ball, when it contacts the ground,
h - height of ball above the ground,
h0 - initial height of ball above the ground.
Height h is given by the formula: h=h0−gt22 We need to find parameters: tend and Vend. Condition for tend is: 0=h0−gt2end2 Finally, tend is: tend=√2h0g Condition for Vend is: mgh0=mV2end2 Finally, Vend is: Vend=√2gh0
Ad 2. Horizontal projection
Below picture (fig 2) presents this type of motion:
Fig 2.
Ball is moving in gravitational field according to horizontal projection motion, when its:- initial position is above the ground level,
- has initial horizontal velocity V0.
→V0=[V0,0] - initial horizontal velocity, →Vg=−gt - velocity caused by gravitational acceleration.
Parameters, that describe horizontal projection:
t0 - time when horizontal projection starts,
tend - time, when ball falls to the ground (moment of contact with Earth's surface),
V - ball's speed,
Vend - speed of ball, when it contacts the ground,
h - height of ball above the ground,
h0 - initial height of ball above the ground,
d - distance that ball moves.
Height h is given by the same formula as free fall (equation (7)).
We need to find parameters: tend, V, Vend, d.
tend is actually identical with tend for free fall motion (given by equation (8)) Based on formula (10), we can calculate speed of ball V: V=|→V|=√V20+(gt)2 Condition for Vend is energy conservation law: mgh0+mv202=mV2end2 Finally, after calculations we get: Vend=√V20+2gh0 Distance d that balls reaches is pretty easy to figure out. It is just V0tend: d=V0√2h0g..
Ad 3. Angular projection
Below Fig 3. presents this type of ball's motion:
Fig 3.
Parameters, that describe angular projection are:t0 - time when angular projection starts,
tmax - time after which ball reaches hmax,
tend - time, when ball falls to the ground (moment of contact with ground level),
V - ball's speed,
→V0 - initial ball's velocity vector,
V0 - initial speed of ball,
α - angle between initial velocity vector →V0 and ground level,
Vend - speed of ball, when it contacts the ground (in t=tend moment),
h - height of ball above the ground,
h0 - initial height of ball laying on the ground. It equals to its radius r. If r≪hmax, then h0≈0,
hmax - max height of ball above the ground,
d - distance that ball moves.
Initial velocity vector →V0 can be present in general form: →V0=[V0x,V0y]. We can present initial velocity vector's components by α angle and its length V0 as follows: {V0x=V0⋅cos(α)V0y=V0⋅sin(α) Therefore, height of ball h is: h=V0⋅sin(α)⏟V0y⋅t−gt22 To calculate hmax we use as usual energy conversation law: mghmax+mV20x⏞V20cos2(α)2=mV202 After basic calculations, we get: hmax=V20y⏞V20⋅sin2(α)2g To calculate Vend we use energy conversation law again: mV202=mV2end2. Based on above equation, we can easily conclude that: Vend=V0. To calculate tmax we use equation (15) to create proper condition: hmax=V0ytmax−gt2max2. hmax is known because we calculate it before. It can be put to above equation: V0y22g=V0ytmax−gt2max2. Finally, after basic calculations, tmax is: tmax=V0yg. Based on equation (15), we can write condition for tend parameter: 0=V0y⋅tend−g⋅t2end2. After simple transformations we get: tend=2V0yg. We can easily figure out that tend=2⋅tmax, which means that angular projection motion is completely symmetrical.
Distance of ball d is given by similar equation to eq(13): d=V0x⋅tend=V0cos(α)⋅2V0yg. After simple calculations we get final version: d=V20gsin(2α).