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4-5 Projectile Motion: Key Characteristics

The above equations for projectile motion can be used to derive several important properties of the motion. The key characteristics and symmetries are

• The path, or trajectory, that a projectile follows is a parabola.
  


• If the initial and final elevations are the same, the range (R) of a projectile, which is the horizontal distance it travels before landing, is given by
  

  .

  
  
• If the initial and final elevations are the same, the launch angle that produces maximum range is q = 45^o.
  


• If the initial and final elevations are the same, the amount of time the projectile spends in the air, sometimes called the time-of-flight, is given by
  

  .

  
  
• If the initial and final elevations are the same, the time it takes a projectile, launched at some upward angle, to reach its maximum height equals the time it takes to fall from its maximum height back down.
  


• The maximum height that a projectile will go above its initial elevation is given by
  

  .

  
  
• The speed that a projectile has at a given height, on its way up, is equal to the speed it will have at that same height on its way back down.
  


• At a given height the angle of the velocity of a projectile above the horizontal, on its way up, equals the angle of the velocity below the horizontal on its way down.

Notice that all of these characteristics are determined by the initial velocity given to the projectile.

Physlet Illustration: The Range of a Projectile
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q = °
A cannon ball is fired from the ground at an initial velocity of 20 m/s.  You control the angle (between 5° and 85°) at which it is launched.  How does the range depend upon the angle?  In the absence of air resistance, what angle gives the maximum range?
Hints Measure the range for several angles less than 50°. Measure the range for an angle less than 45° and one greater than 45°. Try one at 45°. Can you hit the red X?

Physlet Illustration: Shooting Over A Mountain
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v = m/s q = °  Position of Boat = m
Your artillery battalion is stationed on the inland side of a mountain range.  Your commander orders you to fire on ships that pass by just off-shore.  In order to conserve munitions, you know that you should calculate the minimum and maximum angles that the projectile can have in order to make it over the mountains. You control both the angle and speed (5 m/s < v < 100 m/s) of the projectiles. What range of distances off-shore can you hit?  Are there any places where ships are safe to travel?
Hints For a given speed, what minimum angle must the projectile have in order to make it over the mountain?   Is there a maximum angle?  Why? What is the range of a projectile fired at a particular velocity?

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Exercise 4.5 At the Driving Range: A golf ball sitting on level ground i struck and given an initial velocity of 41.2 m/s at an angle of 58.0^o. (a) How high does the ball go into the air? (b) How far does it travel? (c) How long is the ball in the air?

Solution Try to sketch a picture for this problem; the ball moves in a parabolic path starting and ending on the ground. The following information is supplied in the problem

Given: v₀ = 41.2 m/s, q = 58.0^o Find: (a) y_max, (b) R, (c) t

We are given the initial velocity and we know that it completely specifies the motion. Making use of the known results we can directly solve for each of these quantities.

(a)

(b)

(c)

The questions asked in this problem are some of the basic things you might want to know about a projectile. Hopefully, this illustrates the utility of working out equations for certain quantities once and for all.

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Practice Quiz

In general, what is the shape of the path of a projectile? a hyperbola a parabola a straight line a circle

A projectile is launched from the ground with an initial velocity of 44.0 m/s at a launch angle of 26.0o. How long will this projectile be in the air? 8.97 s 8.06 1.97 s 3.93 s

A projectile reaches a height h when launched at an angle q with speed v0. If this projectile is launched from the same level at the same angle with speed 2v0, how high will it go? h 2h 4h h/2

A projectile remains airborne for a time t when launched at an angle q with a speed v0. If this projectile is launched from the same level at the same angle with speed 2v0, how long will it he airborne? t 2t 4t t/2

A projectile travels a distance R when launched at an angle q with speed v0. If this projectile is launched from the same level at the same angle with speed 2v0, how far will it go? R 2R 4R R/2

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