Question

A projectile is launched on the Earth with a certain initial velocity and moves without air resistance. Another projectile is launched with the same initial velocity on the Moon, where the acceleration due to gravity is one-sixth as large. How does the maximum altitude of the projectile on the Moon compare with that of the projectile on the Earth?

(a) It is one-sixth as large.

(b) It is the same.

(c) It is$\sqrt{\mathbf{6}}$ times larger.

(d) It is 6 times larger.

(e) It is 36 times larger.

Short answer

The projectile's altitude on the Moon is six times greater than it is on Earth.

The option (d) is correct.

Step-by-step solution

Step 1: Define maximum height of the projectile

The highest point reached along the object's flight determines its maximum height. The projectile's range is determined by the object's starting velocity and the angle at which it is released.

Explanation for correct option

Maximum altitude that the projectile can reach is-

$h=\frac{v_i^2{\sin }^2{q}_i}{2g}$

The projectile's initial velocity is$v_i$ , the projectile's angle above the horizontal is ${\theta }_i$and the acceleration due to gravity is g.

On the surface of the Earth, a projectile is fired with a high beginning velocity. On the Moon's surface, another projectile is fired with the same initial velocity.

The projectiles are fired at same angle, both on moon and earth. Because the projectiles' initial velocity and projectile angle are the same, their maximum altitude is inversely proportional to their acceleration due to gravity.

$h\propto \frac{1}{g}\dots \dots .\left(1\right)$

Therefore, as, the acceleration due to gravity $\left(g_{\text{M}}\right)$on the Moon is equal to the one sixth of the acceleration due to gravity$\left(g_E\right)$ on the Earth, the maximum height attained by projectile on the moon will be six times that on earth.

So, correct option is d.