Question

A particle is projected at an angle alpha to the horizontal, up and down in a plane, inclined at an angle beta to the horizontal. If the ratio of time of of flights be 1:2, then the ratio tan alpha/tan beta is equal to ?​

Answer 1

Answer:

A particle is projected up an inclined plane of inclination β at an elevation a to the horizon. Show that

a. tanα+cotβ+2tanβ, if the particle strikes the plane at right angles

b. tanα=2tanβ if the particle strikes the plane horizontally

Answer 2

Given:

Angle of projection = α

Angle of inclination = β

Ratio of time of of flights = 1 : 2.

To Find:

The ratio tan α/tan β is to be found.

Solution:

Using the equation for time of flight and the given ratio, we get,

$\frac{\frac{2.U_{o}.Sin (\alpha -\beta ) }{g.Cos \beta }}{\frac{2.U_{o}.Sin (\alpha + \beta ) }{g. Cos \beta } } = T = \frac{1}{2}$

On simplifying, the above equation becomes,

$\frac{Sin (\alpha - \beta )}{Sin (\alpha + \beta )} = \frac{1}{2}$

Using the trigonometric identities for $Sin (A - B)$ and $Sin (A + B)$, the above equation becomes,

$\frac{Sin \alpha. Cos \beta - Cos \alpha. Sin \beta }{Sin \alpha. Cos \beta + Cos \alpha. Sin \beta } = \frac{1}{2}$

Cross multiply the above equation to get,

$2.Sin \alpha. Cos \beta - 2.Cos \alpha. Sin \beta = Sin \alpha. Cos \beta + Cos \alpha. Sin \beta$

On simplifying, we get,

$Sin \alpha. Cos \beta = 3. Cos \alpha. Sin \beta$

Rearranging the above equation;

$\frac{Sin \alpha }{Cos \alpha } = 3. \frac{Sin \beta }{Cos \beta } \\\\i.e., tan \alpha = 3. tan \beta$

$\frac{tan \alpha }{tan \beta } = 3.$

Hence, if the ratio of time of flights is 1 : 2, then the ratio tan α/tan β is equal to 3.

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