Ballistic trajectory
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates the trajectory towards a target in real-time, disregarding air resistance, of a projectile with a given initial velocity v0. The animation does not use formulas (no equations or differential calculus); it only makes the necessary variations in the vectors that direct the motion.
At point A is a cannon and at point B is the target. You can move both points. This animation follows the same steps as the activity "Parabolic motion", except now the initial value v0 of the velocity vector v is introduced in two stages: first its magnitude (which corresponds to the speed of the cannon used) and then its direction (that is, the cannon is aimed to hit the target B).
Move the green point, trying to estimate the correct direction, and then press the button 
. If you do not want to wait for the travel time, activate the Theoretical arc box to see the trajectory that the projectile will follow.
Unless point B is positioned exactly at the limit of the cannon's range, there are either two possible firing angles or none. In the first case, you can see both trajectories by activating the Arches for target box. In the second case, B is outside the cannon's range, and a message will appear to that effect.
. If you do not want to wait for the travel time, activate the Theoretical arc box to see the trajectory that the projectile will follow.
Unless point B is positioned exactly at the limit of the cannon's range, there are either two possible firing angles or none. In the first case, you can see both trajectories by activating the Arches for target box. In the second case, B is outside the cannon's range, and a message will appear to that effect.SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M
SetValue(v, v + dt g)
SetValue(M, If(y(M + dt v)>0, M + dt v, Intersect(Line(M, M + v), xAxis)))
# Adds the position M to the record for the polyline trace and controls the end
SetValue(reg, Append(reg, M))
StartAnimation(anima, y(M) > 0 ∧ abs(M − B) > abs(A − B)/100)
Author of the activity and GeoGebra construction: Rafael Losada.