Name: Inertial Frames of Reference
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Imagine if you are on a bus or a train, can you say you are in an inertial frame of reference? If yes, how can you predict it? Let us explore more in this vi...

Question

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time

t

is proportional to

t^{2}

.

Accepted Answer

Let the initial velocity of the first particle be $u_{1}$ and the angle of projection be $θ_{1}$ with the horizontal. Hence, the components of velocity in the horizontal and vertical direction:

${(u_{1})}_{x} = u_{1} \cos θ_{1}$

${(u_{1})}_{y} = u_{1} \sin θ_{1}$

In the vertical direction, the motion is accelerated in the vertically downward direction while it is uniform in the horizontal direction. After a time interval $t$ , the displacement in the horizontal and vertical direction:

$x_{1} = {(u_{1})}_{x} t$ $= u_{1} t \cos θ_{1}$

$y_{1} = {(u_{1})}_{y} t - \frac{1}{2} g t^{2} = u_{1} t \sin θ_{1} t - \frac{1}{2} g t^{2}$

If the coordinates of initial point are $(0, 0)$ , then the coordinates of the first particle will be $(x_{1}, y_{1})$ .

Similarly, for the other two particles, the position coordinates after a time interval of $t$ will be $(x_{2}, y_{2})$ and $(x_{3,} y_{3})$ where:

$x_{2} = u_{2} t \cos θ_{2}$

$y_{2} = u_{2} t \sin θ_{2} - \frac{1}{2} g t^{2}$

$x_{3} = u_{3} t \cos θ_{3}$

$y_{3} = u_{3} t \sin θ_{3} - \frac{1}{2} g t^{2}$

The area of the triangle formed by these coordinates:

$∆ = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]$

On putting the values of position coordinates of the particles we get:

$∆ = \frac{1}{2} [u_{1} t \cos θ_{1} (u_{2} t \sin θ_{2} - u_{3} t \sin θ_{3}) + u_{2} t \cos θ_{2} (u_{3} t \sin θ_{2} - u_{1} t \sin θ_{3}) + u_{3} t \cos θ_{3} (u_{1} t \sin θ_{1} - u_{2} t \sin θ_{2})]$

$∆ = \frac{t^{2}}{2} [u_{1} \cos θ_{1} (u_{2} \sin θ_{2} - u_{3} \sin θ_{3}) + u_{2} \cos θ_{2} (u_{3} \sin θ_{2} - u_{1} \sin θ_{3}) + u_{3} \cos θ_{3} (u_{1} \sin θ_{1} - u_{2} \sin θ_{2})]$

$\Rightarrow ∆ \propto t^{2}$ (All other terms being constant)

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time $t$ is proportional to $t^{2}$ .

Important Questions on Motion of a Particle

Learn and Prepare for any exam you want !

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time t is proportional to t2.

Important Questions on Motion of a Particle

Learn and Prepare for any exam you want !

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time $t$ is proportional to $t^{2}$ .