Question

Two particles are projected with same speed in different directions in the same vertical plane.both particles have same range .if their time of flights are 10 seconds and 25 seconds then range is (g=10 m/s^2)???

Answer 1

Let the both particles  are projected   with velocity of  v  and  and  angle  of α and β respectively,

For, First body,

$R_1=\frac{u^{2}\sin2\alpha}{g}\\ \\ T_1=\frac{2u\sin\alpha}{g}.........(1)$

For  second body,

$R_2=\frac{u^{2}\sin2\beta}{g}\\ \\ T_2=\frac{2u\sin\beta}{g}.........(2)$

According  to question,

$R_1=R_2\\ \\u^{2}\sin2\alpha}{g}=u^{2}\sin2\beta}{g}\\ \\\sin2\alpha=\sin2\beta\\ \\\sin2\alpha=\sin(180-2\beta)\\ \\2\alpha=180-2\beta\\ \\ \alpha=90-\beta$

T₁/T₂ = sinα/sinβ

10/25 = sinα/sin(90-α)

2/5 = sinα/cosα

tanα = 2/5

sinα = 2/√29

cosα = 5/√29

T₁ = 2usinα/g

$10=\frac{2u\frac{2}{\sqrt{29}}}{10}\\ \\u=\frac{25}{\sqrt{29}}\\ \\ R_1=R_2=\frac{u^{2}\sin2\alpha}{g}\\ \\=\frac{25\times25\times2\sin\alpha.\cos\alpha}{29\times10}=\frac{25\times25\times2\frac{2}{\sqrt{29}}.\frac{5}{\sqrt{29}}}{29\times10}=1250\;m$