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maximizes the range where @p is the solution of the transcendental equatior An exact analytic expression for the launch angle that maximizes the range can be obtained for a special case. The equations of motion given by equations (1) and (2) can be written in the form These are the equations of motion of a wind-free projectile resisted linearly in a frame rotate by an amount / but with an effective gravity of magnitude g’. By comparing equations (1) an (2) with equations (21) and (22) we obtain

Figure 5 maximizes the range where @p is the solution of the transcendental equatior An exact analytic expression for the launch angle that maximizes the range can be obtained for a special case. The equations of motion given by equations (1) and (2) can be written in the form These are the equations of motion of a wind-free projectile resisted linearly in a frame rotate by an amount / but with an effective gravity of magnitude g’. By comparing equations (1) an (2) with equations (21) and (22) we obtain

Related Figures (6)

The physical meaning of @¢ will be discussed in section 5. One feature of equation (6) is that it uses not only the principal branch Wo but also the W_, branch. On physical grounds it turns out that the path the projectile takes before it reverses its horizontal motion uses W_, and the remaining part of the path uses Wo. The criteria for reversing the horizontal motion before landing will be discussed further in section 4. be the principal branch, usually denoted by W (x), while W(x) < —1 is denoted by W_,(x). Using equations (3) and (4) the shape of the trajectory can be expressed using the Lambert W function:
The physical meaning of @¢ will be discussed in section 5. One feature of equation (6) is that it uses not only the principal branch Wo but also the W_, branch. On physical grounds it turns out that the path the projectile takes before it reverses its horizontal motion uses W_, and the remaining part of the path uses Wo. The criteria for reversing the horizontal motion before landing will be discussed further in section 4. be the principal branch, usually denoted by W (x), while W(x) < —1 is denoted by W_,(x). Using equations (3) and (4) the shape of the trajectory can be expressed using the Lambert W function:
4. Range versus maximum horizontal displacement For wind-influenced projectiles, the range may not be equal to the maximum horizontal displacement. By setting the horizontal component of velocity to zero, using equation (3), and solving for the time, then we get an expression for the time tp at which the projectile reaches its maximum horizontal displacement
4. Range versus maximum horizontal displacement For wind-influenced projectiles, the range may not be equal to the maximum horizontal displacement. By setting the horizontal component of velocity to zero, using equation (3), and solving for the time, then we get an expression for the time tp at which the projectile reaches its maximum horizontal displacement
then the range would not be equal to the maximum horizontal displacement. where p is the solution of the transcendental equation 5. Critical angle: forward and backward landing The range will not be equal to the maximum horizontal displacement when the projectile reverses its horizontal motion before landing, tp < tight. Using equations (9) and (13), we see that if the launch angle @ satisfies
then the range would not be equal to the maximum horizontal displacement. where p is the solution of the transcendental equation 5. Critical angle: forward and backward landing The range will not be equal to the maximum horizontal displacement when the projectile reverses its horizontal motion before landing, tp < tight. Using equations (9) and (13), we see that if the launch angle @ satisfies
Figure 1. Trajectories for projectiles with launch angles 6 = 0.96¢, Oc, 1.16¢ with 7 = 0.981 and U = > for a = 180°. kvo
Figure 1. Trajectories for projectiles with launch angles 6 = 0.96¢, Oc, 1.16¢ with 7 = 0.981 and U = > for a = 180°. kvo
which the range goes to zero is given by 0c. Backward displacement is maximized by 6 = 90° when it exists.
which the range goes to zero is given by 0c. Backward displacement is maximized by 6 = 90° when it exists.
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Projectile Motion
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