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a projectile can have the same range R for two angles of projection.if t 1 and t 2 be the times of flight in the two cases,then what is the product of two times of flight?

a projectile can have the same range R for two angles of projection.if t1 and t2 be the times of flight in the two cases,then what is the product of two times of flight?

Grade:11

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
9 years ago
Dear student,
We have,
\frac{u_{1}^{2}sin\left ( 2\theta _{1} \right )}{g}=\frac{u_{2}^{2}sin\left ( 2\theta _{2} \right )}{g} ... (i)
Now,
R^{2}=\frac{u_{1}^{2}sin\left ( 2\theta _{1} \right )}{g}\times \frac{u_{2}^{2}sin\left ( 2\theta _{2} \right )}{g}=\frac{4\left ( u_{1}u_{2} \right )^{2}sin\left ( \theta _{1} \right )sin\left ( \theta _{2} \right )cos\left ( \theta _{1} \right )cos\left ( \theta _{2} \right )}{g^{2}}This, on solving gives:
\frac{R^{2}}{u_{1}u_{2}cos\left ( \theta _{1} \right )cos\left ( \theta _{2} \right )}=t_{1}t_{2}
Substituting for the speeds and the angles from the first equation, would give the required result.
Regards
Sumit

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