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Projectile Trajectory Computer Simulation Launch Angle 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Launch Speed 10 m/s (22 mph) human runner 40 m/s (90 mph) thrown baseball 50 m/s (112 mph) arrow, tennis ball 75 m/s (168 mph) golf ball 150 m/s (336 mph) 60mm mortar 300 m/s (671 mph) pistol bullet 500 m/s (1,119 mph) 155mm artillery 800 m/s (1,790 mph) 7.62 mm NATO bullet 1000 m/s (2,237 mph) M16 bullet, 88mm anti-aircraft shell Simulation Speed Real Time Speeded Up Gravity Earth (9.8 meters/second/second) Mars(3.7 meters/second/second) Moon (1.6 meters/second/second)                           Home

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Flight Time = . seconds Vertical Height = . meters Horizontal Distance = . meters

Maximum Height = . meters Horizontal Speed = . meters / second Vertical Speed = . meters / second

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Projectiles are objects that are thrown (or shot, hit etc) into the air, then left to move under their own speed, subject to the influence of gravity.

Unlike rockets and aircraft, projectiles have only their starting speed to power their flight.

Notes

• Simulation works in real time, reflecting actual elapsed time.
• Display is automatically scaled to fit the projectile's entire trajectory. Changing the launch speed will change the scale. Pay attention to the scale at the bottom of the display. Obviously, the size of the projectile itself is not scaled with the display, and will always be shown at a fixed size.
• The "tail" of the projectile is to help visualize the trajectory. It's length does not indicate anything.

Useful Numbers

• "Outer space" is commonly defined as beginning 100 kilometers above the surface of the Earth.
• Diameter of the Earth is 12,700 kilometers (12,700,000 m).
• Escape velocity from the surface of the Earth is 11,200 meters/sec, but no matter how high the launch speed in the simulation, the projectile will always return to Earth - see Simulation Limitations below.
• Speed of sound is 344 meters/sec (actual value depends on air temperature and density)

Exercises

1. For a given launch speed, what angle gives the

• furthest horizontal distance
• highest vertical height
• fastest horizontal speed
• fastest vertical speed
• longest flight time

and why.

2. Do the angles above, change depending on the launch speed, or are they always the same?

3. Plot different launch speeds against the values in 1. What is the relationship?

4. Compare the world athletic records for high jump and long jump. How close or far are they from the theoretical maximum height and distance for a 10 m/s projectile? Why (hint: not because of aerodynamics)?

5. Artillery and mortars are used at or close to their actual (real) maximum ranges, but pistols and rifles are not. Why?

Simulation Limitations

• Assumes uniform gravitational acceleration. This makes the simulations less accurate for high launch speeds and large launch angles (the higher the projectile, the less the gravitational acceleration).
• Assumes zero aerodynamic drag. This makes the simulation less accurate for high launch speeds (the higher the speed, the higher the aerodynamic drag) and small projectiles (the smaller the projectile, the larger the surface area / mass volume, the larger the aerodynamic drag relative to the kinetic energy - an M16 bullet and an 88mm shell have the same muzzle velocity, but the 88mm shell has a far greater range).

How it Works

Projectile trajectory is calculated numerically using the formula

s = (u * t) + (0.5 * a * t * t)

where

• s = distance travelled
• u = launch speed
• t = time
• a = acceleration

The program loops with increasing values of t, calculating the value of s at each instance of t. Separate calculations are made for vertical and horizontal distances. u is multiplied with sine and cosine to separate out the vertical and horizontal components of the launch speed.

Speed at any instance is approximated by comparing the current s with the previous s, and dividing by the t interval, v = (s2 - s1) / (t2 - t1) . The formula v² = u² + 2as could also have been used.

Background

Military use of artillery (the firing of projectiles) has often given practical (computational) mathematics a boost.

The standard A, B, C and D scales of the slide-rule (a simple analog computer or calculator) were created by the French artillery officer Mannheim in 1859, to calculate firing solutions for his guns.

During World War II, American electro-mechanical (electrical relay) digital computers were developed to compute artillery firing tables (for accurate results, air density - altitude, temperature - were added to the calculations).