Understanding Projectile Motion : Derivations

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Projectile Motion is one of the basic thing we encounter in our daily life. Studying it becomes important because the ‘understanding’ (not mugging up Formulae) which we take here comes handy while analyzing complex concepts and experiments 

“Formulae give you Marks but Derivations give you Understanding !

We already do have a video on our Channel about such a experiment on our channel where knowledge of Projectile is needed for the Analysis part :

Equation of Trajectory

We all have played ‘catch-catch’ and we know how the motion of a ball when thrown looks like. But how do you describe that curve mathematically. What’s the equation of that curve ?

Most Important Tip that fixes everything related to Projectile !

Divide this whole 2-D situation into separate 1-D problems (X and Y)

  • Note down all quantities in X separately. This will be your separate problem
  • Note down all quantities in Y separately. This will be your another separate problem
  • At last, combine them to get results for 2-D motion

Following the Tip :

Starting from origin (point where ball is throwed) till point P (x,y) :

The Projectile Trajectory is a Parabola !

Finding Range Expression

Range is basically the ‘horizontal’ distance which the ball covers (from origin to the point where it lands). It means that, we need to find R in the figure

  • This can be easily found since x = 0 and x = R are the two roots of the parabola.
  • To calculate the roots, simply put y = 0 in our Equation of Trajectory

Calculations :

Substituting y = 0 and taking ‘x’ common on RHS of Equation of Trajectory, we get : 

* In the simplifying process, you need to use sin2theta trigonometric identity

Finding Time of Flight Expression

Let’s find out for how much time does the body stay in the projectile motion.

Note that :

  • On landing, after completing the motion, the displacement in Y direction is zero (Pause and observe !), since it again came to the same Y-level (y = 0 here)
  • The time taken for this displacement in Y to become 0 is nothing but the Time of Flight (T)

Finding Maximum Height Covered :

If we look at the Equation of Continuity, in this case (simple throwing); the trajectory resembles downward parabola since a<0

Hence visualizing it graphically, 

Finishing the Calculations,

So, this is how, we get the expression for the Maximum Height reached, denoted by ‘H’

Conclusion

So, this is how we derive the expressions for Time of Flight, Range, Maximum Height and Equation of Trajectory. Note that – With this same approach, we can solve almost any kind of problem related to Projectile Motion. And, Why is that the case?

Because, whatever we discussed in this blog/article is not a trick or something, it’s a complete concept with complete understanding. This makes us equipped to solve any kind of problem (Throwing from a cliff, projectile on incline or just anything…)

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