Artillery A projectile fired into the first quadrant from the origin of a coordinate system will pass through the point ( x , y ) at time t according to the relationship cot θ = 2 x 2 y + g t 2 where θ = the angle of elevation of the launcher and g = the acceleration due to gravity = 32.2 f e e t / s e c o n d 2 . An artilleryman is firing at an enemy bunker located 2450 feet up the side of a hill that is 6175 feet away. He fires a round, and exactly 2.27 seconds later he scores a direct hit. (a) What angle of elevation did he use? (b) If the angle of elevation is also given by sec θ = v 0 t x , where v 0 is the muzzle velocity of the weapon, find the muzzle velocity of the artillery piece he used.
Artillery A projectile fired into the first quadrant from the origin of a coordinate system will pass through the point ( x , y ) at time t according to the relationship cot θ = 2 x 2 y + g t 2 where θ = the angle of elevation of the launcher and g = the acceleration due to gravity = 32.2 f e e t / s e c o n d 2 . An artilleryman is firing at an enemy bunker located 2450 feet up the side of a hill that is 6175 feet away. He fires a round, and exactly 2.27 seconds later he scores a direct hit. (a) What angle of elevation did he use? (b) If the angle of elevation is also given by sec θ = v 0 t x , where v 0 is the muzzle velocity of the weapon, find the muzzle velocity of the artillery piece he used.
Solution Summary: The author calculates the angle of elevation used by the artillery man to hit the projectile on the enemy bunker.
Artillery A projectile fired into the first quadrant from the origin of a coordinate system will pass through the point
at time
according to the relationship
where
the angle of elevation of the launcher and
the acceleration due to gravity
. An artilleryman is firing at an enemy bunker located 2450 feet up the side of a hill that is 6175 feet away. He fires a round, and exactly
seconds later he scores a direct hit.
(a) What angle of elevation did he use?
(b) If the angle of elevation is also given by
, where
is the muzzle velocity of the weapon, find the muzzle velocity of the artillery piece he used.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Expert Solution
To determine
a. The angle of elevation used by the artillery man to hit the projectile on the enemy bunker.
Answer to Problem 81AYU
Solution:
The angle of elevation used by the artillery man to hit the projectile on the enemy bunker.
Explanation of Solution
Given:
The height of the enemy bunker .
Horizontal location of the enemy bunker from the projectile artillery man .
Time taken for the projectile to hit the enemy bunker .
acceleration due to gravity .
Formula used:
angle of elevation of the launcher.
Horizontal location of the enemy bunker from the projectile artillery man (Launcher).
The height of the enemy bunker.
Time taken for the projectile to hit the enemy bunker.
acceleration due to gravity.
Calculation:
To find , we need to find because has the same range as except where undefined.
Equation of the circle .
Where .
Both and are positive, therefore lies in the I quadrant.
Using the formula,
Expert Solution
To determine
b.The muzzle velocity of the artillery piece.
Answer to Problem 81AYU
Solution:
The muzzle velocity of the artillery piece .
Explanation of Solution
Given:
Horizontal location of the enemy bunker from the projectile artillery man .
Time taken for the projectile to hit the enemy bunker .
Thomas' Calculus: Early Transcendentals (14th Edition)
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