Question

A projectile is fired from the origin of the coordinate system with an initial velocity v0 in a direction making an angle α with the horizontal. Calculate the time required for the projectile to cross a line passing through the origin and making an angle β < α with the horizontal.

Answer 1

Answer:

Explanation:

Given

launching inclination is $\alpha$

suppose u is the initial velocity

vertical velocity $v_y=u\sin \alpha$

horizontal velocity $v_x=u\cos \alpha$

Vertical distance traveled by Projectile is

$y=v_y\times t-\frac{1}{2}gt^2----1$

Now equation of line passing through origin

$y=x\tan \beta------2$

Equating two equation we get

$x\tan \beta=v_y\times t-\frac{1}{2}gt^2$

putting $x=v_x\times t=u\cos \alpha \times t$

$u\cos \alpha \tan \beta \cdot t=v_y\times t-\frac{1}{2}gt^2$

$t\left [ u\sin \alpha -u\cos \alpha \tan\beta -\frac{gt}{2}\right ]=0$

$t=\frac{2u\left [ \sin \alpha -\cos \alpha \tan \beta \right ]}{g}$