Projection Under uniform Acceleration

Question

A projectile is fired from the surface of the earth with a velocity of $5m{s}^{-1}$and angle $\theta $ with the horizontal. Another projectile fired from another planet with a velocity of $3m{s}^{-1}$ at the same angle follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the planet is $\left(m{s}^{-2}\right)$ is (Given g = $m{s}^{-2}$

Moderate

Solution

The equation of trajectory is

$y=x\mathrm{tan}\theta -\frac{g{x}^{2}}{2{u}^{2}{\mathrm{cos}}^{2}\theta}$

where $\theta $ is the angle of projection and u is the velocity with which projectile is projected. For equal trajectories for same angles of projection,

$\frac{g}{{u}^{2}}=\text{constant}$

As per question $\frac{9.8}{{5}^{2}}=\frac{{g}^{\mathrm{\prime}}}{{3}^{2}}$

where g' is acceleration due to the gravity on the planet.

${g}^{\mathrm{\prime}}=\frac{9.8\times 9}{25}=3.5{\mathrm{ms}}^{-2}$

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