Motion in Two Dimension
6.0 Relative motion between two projectiles
6.0 Relative motion between two projectiles
Consider a particle is projected with velocity $u_1$ at an angle $\theta _1$ with the horizontal. At the same time, another particle is projected with velocity $u_2$ at an angle $\theta _2$ with the horizontal as shown in the figure.
We will write the kinematic variables of each particle in vector form.
| Kinematic variable | Particle $A$ | Particle $B$ |
|
| |
| Velocity | ${\overrightarrow u _A} = {u_1}\left( {\cos {\theta _1}\widehat i + \sin {\theta _1}\widehat j} \right)$ | ${\overrightarrow u _B} = {u_2}\left( {\cos {\theta _2}\widehat i + \sin {\theta _2}\widehat j} \right)$ |
| Acceleration | ${\overrightarrow a _A} = - g\widehat j$ | ${\overrightarrow a _B} = - g\widehat j$ |
Let us assume body $B$ to be the observer.
So, $${\overrightarrow u _{AB}} = {\overrightarrow u _A} - {\overrightarrow u _B}$$$${\overrightarrow u _{AB}} = {u_1}\left( {\cos {\theta _1}\widehat i + \sin {\theta _1}\widehat j} \right) - {u_2}\left( {\cos {\theta _2}\widehat i + \sin {\theta _2}\widehat j} \right)$$$${\overrightarrow u _{AB}} = \left( {{u_1}\cos {\theta _1} + {u_2}\cos {\theta _2}} \right)\widehat i + \left( {{u_1}\sin {\theta _1} - {u_2}\sin {\theta _2}} \right)\widehat j$$
Similarly, $${\overrightarrow a _{AB}} = {\overrightarrow a _A} - {\overrightarrow a _B}$$$${\overrightarrow a _{AB}} = - g\widehat j - \left( { - g\widehat j} \right)$$$${\overrightarrow a _{AB}} = 0$$
So, observer $B$ will observe that body $A$ is undergoing motion with uniform velocity.
where,
: Trajectory of the projectile as observed by the observer.is the general concept for relative projectile motion.
Now it has various application such as,
1. Condition for collision
2. Time after which two particles will collide
3. Shortest distance between them
4. Time after which their velocity vector are perpendicular
Note: When the observer is undergoing a projectile motion or it is freely falling under gravity, then the frame of reference is called as a $g$ frame of reference.
In the $g$ frame of reference, relative velocities of all the particles freely falling under gravity or undergoing a projecticle motion is constant.
We will discuss each case with the help of an example.
