A ball is to be shot from level ground with a certain speed v 0 at various angle

Question

A ball is to be shot from level ground with a certain speed v0 at various angles . The launch angle determines both the time of flight t and the minimum speed vmin the ball will have during its flight, which means they can be related to each other. At some angle, the ball will have a maximum possible time of flight. Call this time tmax. Find a formula which gives the ratio of vmin to v0 in terms of t, tmax, and g.

so far I have attempted this problem twice with the answers

both of these answers were incorrect.

Explanation / Answer

initial apeed of the ball   v o       launch angle         time of flight   t =   2 vo sin / g ....(1)     minium speed    v_ minimum = vo sin - gt ....(2)    at = 90 o              Time of flight is maximum                  Tmax   =    2 vo / g ......(3) from equation (3)               vo = g Tmax / 2 .....(4)        vmin / vo   = vo sin - gt     / ( g Tmax / 2)                              = 2 vo sin / g Tmax     -   t /   T Max                at maximum height seed of the ball = 0              velocity of the ball before reach the maximum height   = Vmin         t = time taken by the ball to reach the point at which its velocity is vmin initial apeed of the ball   v o       launch angle         time of flight   t =   2 vo sin / g ....(1)     minium speed    v_ minimum = vo sin - gt ....(2)    at = 90 o              Time of flight is maximum                  Tmax   =    2 vo / g ......(3) from equation (3)               vo = g Tmax / 2 .....(4)        vmin / vo   = vo sin - gt     / ( g Tmax / 2)                              = 2 vo sin / g Tmax     -   t /   T Max                at maximum height seed of the ball = 0              velocity of the ball before reach the maximum height   = Vmin         t = time taken by the ball to reach the point at which its velocity is vmin initial apeed of the ball   v o       launch angle         time of flight   t =   2 vo sin / g ....(1)     minium speed    v_ minimum = vo sin - gt ....(2)    at = 90 o              Time of flight is maximum                  Tmax   =    2 vo / g ......(3) from equation (3)               vo = g Tmax / 2 .....(4)        vmin / vo   = vo sin - gt     / ( g Tmax / 2)                              = 2 vo sin / g Tmax     -   t /   T Max                at maximum height seed of the ball = 0              velocity of the ball before reach the maximum height   = Vmin         t = time taken by the ball to reach the point at which its velocity is vmin

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