EXAMPLES 215 



5. A shot fired from a gun whose muzzle is close to the ground just clears a 

 man 6 feet high standing 10 yards away, and embeds itself in the ground a 

 quarter of a mile off. Show that the shot rises to a height above the ground 

 which is certainly greater than 22 yards. 



6. A projectile has a maximum horizontal range of 256 feet ; what is its 

 velocity of projection ? 



If it be projected with this velocity from a point on the floor of a long 

 corridor 24 feet high, what will be its greatest range, if it is not to strike the 

 ceiling ? 



7. Prove that the velocity required for a range of 20 miles will be at least 

 1840 feet per second, with a time of flight of 81.3 seconds. 



8. Determine the charge of powder required for the range of 20 miles in the 

 last question, supposing the shot to weigh a ton, and the strength of the powder 

 capable of realizing 100 foot-tons per pound of powder. 



9. Show that the range E of a projectile fired from a height h above a level 

 plane with velocity v at an angle a is given by 



2 u 2 (h + R tan a) = gE 2 sec 2 a. 



10. Show that the area of a level plane swept by a gun at a height h above 

 the plane increases proportionally with ft, being equal to 



where A is the area commanded when the gun is at the level of the plane. 



11. A projectile can be fired with a velocity of 1720 feet per second from a 

 fort at a height of 300 feet above a horizontal plane. Find what area of the 

 plane is covered by the gun. 



12. A particle is projected so as just to graze the four upper corners of a 

 regular hexagon of side a, placed vertical with one edge resting on a horizontal 

 table Find the highest point in the flight of the particle, and show that the range 

 on the table is a y?. 



13. A machine-gun is placed on an armored train which runs along a hori- 

 zontal line of rails with velocity v. The muzzle velocity of shots fired from the 

 gun is V. Find the greatest range 



(a) in front of the train ; 

 (6) behind the train. 



GENERAL EXAMPLES 



1. A train is going at 60 miles an hour, when it comes to a curve hav- 

 ing a radius of f of a mile. There is a perfectly smooth horizontal shelf 

 in the train, its edge being parallel to the rails and on the side of the shelf 

 away from the center of the curve. A small object stands on the shelf at 

 a distance of 8 inches from the edge. Show that the object will fall off 

 the shelf after the car containing the object has described about 24 yards 

 of the curve, and find what its horizontal velocity will be when, it leaves 

 the shelf. 



