Projectile Motion is one of the basic phenomenon encountered in our daily life. Studying it becomes important because the ‘understanding’ (not mugging up Formulae) which we take here comes handy in analyzing complex concepts and experiments
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 :
- Calculating e/m Ratio – https://www.youtube.com/watch?v=rxL8DkdYCTw
Table of Contents :
- Equation of Trajectory
- Finding Range expression
- Finding Time of Flight expression
- Finding Maximum Height expression
1. 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 !
2. 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
3. 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)
4. 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 ge the expression for the Maximum Height reached, denoted by ‘H’
