# 4.3 Projectile motion  (Page 7/13)

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Check Your Understanding If the two golf shots in [link] were launched at the same speed, which shot would have the greatest range?

The golf shot at $30\text{°}.$

When we speak of the range of a projectile on level ground, we assume R is very small compared with the circumference of Earth. If, however, the range is large, Earth curves away below the projectile and the acceleration resulting from gravity changes direction along the path. The range is larger than predicted by the range equation given earlier because the projectile has farther to fall than it would on level ground, as shown in [link] , which is based on a drawing in Newton’s Principia . If the initial speed is great enough, the projectile goes into orbit. Earth’s surface drops 5 m every 8000 m. In 1 s an object falls 5 m without air resistance. Thus, if an object is given a horizontal velocity of $8000\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}$ (or $18,000\text{mi}\text{/}\text{hr}\right)$ near Earth’s surface, it will go into orbit around the planet because the surface continuously falls away from the object. This is roughly the speed of the Space Shuttle in a low Earth orbit when it was operational, or any satellite in a low Earth orbit. These and other aspects of orbital motion, such as Earth’s rotation, are covered in greater depth in Gravitation .

At PhET Explorations: Projectile Motion , learn about projectile motion in terms of the launch angle and initial velocity.

## Summary

• Projectile motion is the motion of an object subject only to the acceleration of gravity, where the acceleration is constant, as near the surface of Earth.
• To solve projectile motion problems, we analyze the motion of the projectile in the horizontal and vertical directions using the one-dimensional kinematic equations for x and y .
• The time of flight of a projectile launched with initial vertical velocity ${v}_{0y}$ on an even surface is given by
${T}_{tof}=\frac{2\left({v}_{0}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta \right)}{g}.$

This equation is valid only when the projectile lands at the same elevation from which it was launched.
• The maximum horizontal distance traveled by a projectile is called the range. Again, the equation for range is valid only when the projectile lands at the same elevation from which it was launched.

## Conceptual questions

Answer the following questions for projectile motion on level ground assuming negligible air resistance, with the initial angle being neither $0\text{°}$ nor $90\text{°}:$ (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at t = 0? (d) Can the speed ever be the same as the initial speed at a time other than at t = 0?

a. no; b. minimum at apex of trajectory and maximum at launch and impact; c. no, velocity is a vector; d. yes, where it lands

Answer the following questions for projectile motion on level ground assuming negligible air resistance, with the initial angle being neither $0\text{°}$ nor $90\text{°}:$ (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?

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