3.4 Motion in space  (Page 5/17)

One final question remains: In general, what is the maximum distance a projectile can travel, given its initial speed? To determine this distance, we assume the projectile is fired from ground level and we wish it to return to ground level. In other words, we want to determine an equation for the range. In this case, the equation of projectile motion is

Setting the second component equal to zero and solving for t yields

Therefore, either $t=0$ or $t=\frac{2v_0\text{sin}\theta }{g}.$ We are interested in the second value of t , so we substitute this into $\text{s}\left(t\right),$ which gives

Thus, the expression for the range of a projectile fired at an angle $\theta$ is

The only variable in this expression is $\theta .$ To maximize the distance traveled, take the derivative of the coefficient of i with respect to $\theta$ and set it equal to zero:

This value of $\theta$ is the smallest positive value that makes the derivative equal to zero. Therefore, in the absence of air resistance, the best angle to fire a projectile (to maximize the range) is at a $45\text{°}$ angle. The distance it travels is given by

Therefore, the range for an angle of $45\text{°}$ is $v_0^2\text{/}g.$

Kepler’s laws

During the early 1600s, Johannes Kepler was able to use the amazingly accurate data from his mentor Tycho Brahe to formulate his three laws of planetary motion, now known as Kepler’s laws of planetary motion    . These laws also apply to other objects in the solar system in orbit around the Sun, such as comets (e.g., Halley’s comet) and asteroids. Variations of these laws apply to satellites in orbit around Earth.

Kepler’s third law is especially useful when using appropriate units. In particular, 1 astronomical unit is defined to be the average distance from Earth to the Sun, and is now recognized to be 149,597,870,700 m or, approximately 93,000,000 mi. We therefore write 1 A.U. = 93,000,000 mi. Since the time it takes for Earth to orbit the Sun is 1 year, we use Earth years for units of time. Then, substituting 1 year for the period of Earth and 1 A.U. for the average distance to the Sun, Kepler’s third law can be written as