Question

a particle is projected with a velocity of 20 m/sec at an angle of 30° to an inclined plane of inclination 30° to the horizontal. The time of flight is​

Answer 1

Given:

The particle is projected with an initial velocity, u = 20 m/s

The angle of projection with the inclined plane = 30°

The angle of inclination of the inclined plane = 30°

To find:

The time of flight

Solution:

The formula to find the time of flight, when a particle is projected from an inclined plane is as follows:

$\boxed{\bold{Time\:of\:Flight,\:T\:=\:\frac{2u\:sin\:\alpha}{g\:cos\:\theta} }}$

where

u = initial velocity

α = angle of projection from the inclined plane

θ = angle of inclination

g = acceleration due to gravity = 10 m/s²

Now, we will substitute the given values in the above formula to find the value of T

$T\:=\:\frac{2\:\times\:20\times\:sin\:30}{10\:cos\:30}$

∵ sin 30° = $\frac{1}{2}$ and cos 30° = $\frac{\sqrt{3}}{2}$

⇒ $T\:=\:\frac{2\:\times\:20\times\:\frac{1}{2} }{10\:\times\:\frac{\sqrt{3}}{2} }$

⇒ $T\:=\:\frac{20}{5\:\times\:\sqrt{3}}$

⇒ $T\:=\:\frac{4}{\sqrt{3}}\;sec$

Thus, the time of flight is  $\underline{\bold{{\frac{4}{\sqrt{3}}\;sec}}}$ or 2.3 sec.

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Answer 2

Answer:

Let the horizontal component be x and vertical component be y.

X = v(cos theta)t

Y= v(sin theta)t-1/2 (gt^2)

Initially the slope of the ball to the horizontal =sin theta/cos theta =tan theta

At any time “t” the slope of the ball

=v (cos theta)t/v (sin theta)t-1/2 (gt^2) =dy/dx

If the direction of the ball is perpendicular to the initial direction then:

(dy/dx)*(tan theta) = -1

So,

V(sin^2theta)t-1/2 (gt^2)/V (cos theta )t*(tan theta ) = -1

Hence :-

V (sin^2theta )-(gt*sin theta ) = -(Vcos^2theta )

Thus:-

V (sin^2theta +cos^2theta) =gt (sin theta )

V=gt (sin theta )

So , coming back to the question

Time at which the velocity is perpendicular to the initial velocity is given by T=v/g (sin theta )

Replacing them with numerical values we get:-

T=20/10*sin 30

T=20/10*0.5

T=20/5=4

Therefore after 4 seconds the initialvelocity would be perpendicular to the velocity