Launch and aim angle determination for an object

A method for determining at least one approximate launch parameter of an object in flight after being launched from an initial position comprising the steps of:

• positioning a plurality of sensors for measuring a velocity of the object while in flight, wherein each sensor is positioned along a line substantially perpendicular to a projection onto the ground of a straight flight path expected when the object is launched on a direct bearing towards a desired target location, wherein the line is at a known distance forward of the initial position measured along the projection of the expected straight flight path, and wherein the sensors are positioned along the line at locations corresponding to selected aim angles;
• measuring the velocity of the object while in flight using each of the sensors;
• performing independent launch angle calculations corresponding with each sensor using a mathematical model;
• and
• using the results of the independent launch angle calculations to obtain at least one approximate launch parameter of the object, wherein an approximate launch angle is obtainable by examining the results of the independent launch angle calculations to determine which of the launch angle calculations resulted in the smallest calculated launch angle and selecting the smallest calculated launch angle as the launch angle of the object.

A method for determining a launch angle of a golf ball in flight after being launched from an initial position by striking it with a golf club, said method comprising the steps of:

• operating a sensor positioned in a known geometrical relation with the initial position to obtain a measured velocity of the ball while in flight at an elapsed time after the ball is launched;
• obtaining an actual velocity of the ball after being launched from the initial position;
• and
• calculating the launch angle of the ball in accordance with a mathematical model comprised of the following equations:
• a=d2 (1)
• b=−2dc[1−{vm(t)/vo}2] (2)
• e=[vm(t)/vo2][−c2−d2]+c2 (3)
• root=[b2−4ae]½ (4)
• core=(−b+root)/2a (5)
• θ=arctan[(1−core2)½/core] (6)
• wherein:
• t is the time elapsed since launch of the ball;
• vm(t) is the measured velocity of the ball at a selected time t;
• vo is the actual velocity of the ball;
• c represents the total distance traveled by the ball along a line of sight from its position at time t to the initial position and is computed as the product of vo and t;
• d is the distance between the initial position and the position of the sensor measured along a projection of an expected straight flight path;
• and
• θ is the launch angle of the ball.

An apparatus for use in determining a launch angle of a golf ball in flight after being launched from an initial position by striking it with a golf club, said apparatus comprising:

• a sensor positionable in a known geometrical relation with the initial position and operable to obtain a measured velocity of the ball while in flight at an elapsed time after the ball is launched;
• a mathematical model comprised of the following equations:
• a=d2 (1)
• b=−2dc[1−{vm(t)/vo}2] (2)
• e=[vm(t)/vo2][−c2−d2]+c2 (3)
• root=[b2−4ae]½ (4)
• core=(−b+root)/2a (5)
• θ=arctan[(1−core2)½/core] (6)
• wherein:
• t is the time elapsed since launch of the ball;
• vm(t) is the measured velocity of the ball at a selected time t;
• vo is an the actual velocity of the ball;
• c represents the total distance traveled by the ball along a line of sight from its position at time t to the initial position and is computed as the product of vo and t;
• d is the distance between the initial position and the position of the sensor measured along a projection of an expected straight flight path;
• and
• θ is the launch angle of the ball;
• and
• a computing device operable to calculate the launch angle of the ball in accordance with the mathematical model.

A method for determining at least one launch parameter of a golf ball in flight after being launched from an initial position by striking it with a golf club, said method comprising the steps of:

• operating a plurality of sensors positioned in known geometrical relations with the initial position to obtain a measured velocity of the ball while in flight corresponding with each sensor at an elapsed time after the ball is launched;
• obtaining an actual velocity of the ball after being launched from the initial position;
• performing independent launch angle calculations corresponding with each sensor in accordance with a mathematical model comprising the following equations:
• a=d2 (1)
• b=−2dc[1−{vm(t)/vo}2] (2)
• e=[vm(t)/vo][−c2−d2]+c2 (3)
• root=[b2−4ae]½ (4)
• core=(−b+root)/2a (5)
• θ=arctan[(1−core2)½/core] (6)
• wherein:
• t is the time elapsed since launch of the ball;
• vm(t) is the measured velocity of the ball at a selected time t;
• vo is the actual velocity of the ball;
• c represents the total distance traveled by the ball along a line of sight from its position at time t to the initial position and is computed as the product of vo and t;
• d is the distance between the initial position and the position of the sensor measured along a projection of an expected straight flight path;
• and
• θ is a launch angle of the ball;
• and
• using the results of the independent launch angle calculations to obtain at least one approximate launch parameter of the ball.

An apparatus for use in determining at least one launch parameter of a golf ball in flight after being launched from an initial position by striking it with a golf club, said apparatus comprising:

• a plurality of sensors, each said sensor being positionable in a known geometrical relation with the initial position and operable to obtain a measured velocity of the ball while in flight corresponding with such sensor at an elapsed time after the ball is launched;
• a mathematical model comprising the following equations:
• a=d2 (1)
• b=−2dc[1−{vm(t)/vo}2] (2)
• e=[vm(t)/vo2][−c2−d2]+c2 (3)
• root=[b2−4ae]½ (4)
• core=(−b+root)/2a (5)
• θ=arctan[(1−core2)½/core] (6)
• wherein:
• t is the time elapsed since launch of the ball;
• vm(t) is the measured velocity of the ball at a selected time t;
• vo is an actual velocity of the ball;
• c represents the total distance traveled by the ball along a line of sight from its position at time t to the initial position and is computed as the product of vo and t;
• d is the distance between the initial position and the position of the sensor measured along a projection of an expected straight flight path;
• and
• θ is a launch angle of the ball;
• and
• a computing device operable to perform independent launch angle calculations corresponding with each sensor in accordance with the mathematical model, wherein at least one launch parameter is obtainable from the results of the independent launch angle calculations.