Two particles are projected from the same point

Asked by admin @ 29/08/2021 in Physics viewed by 262 People

Two particle projected from the same point with same speed u at angles of projection α and β strike the horizontal ground at the same point. if h1 and h2 are the maximum heights attained by the projectile, r is the range for both and t1 and t2 are their times of flights, respectively, then

Answered by admin @ 29/08/2021

Given that two particles are projected with same speed and both have same range so we can write

$R_1 = R_2$

$\frac{u^2sin2\alpha}{g} = \frac{u^2sin2\beta}{g}$

$sin2\alpha = sin2\beta$

$2\alpha = 180 - 2\beta[\tex][tex]\alpha + \beta = 90$

now for heights

$h_1 = \frac{u^2 sin^2\alpha}{2g}$

$h_2 = \frac{u^2 sin^2\beta}{2g}$

now the ratio of above two cases

$\frac{h_1}{h_2} = \frac{sin^2\alpha}{sin^2\beta}$

since we know that

$\alpha + \beta = 90$

$\frac{h_1}{h_2} = \frac{sin^2\alpha}{sin^2(90 - \alpha)}$

$\frac{h_1}{h_2} = tan^2\alpha$

now similarly for time of flight

$T_1 = \frac{2usin\alpha}{g}$

$T_2 = \frac{2usin\beta}{g}$

since we know that

$\alpha + \beta = 90$

$T_2 = \frac{2usin(90 - \alpha)}{g}$

$T_2 = \frac{2ucos\beta}{g}$

now the ratio of two cases

$\frac{T_1}{T_2} = tan\alpha$

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