Find the angle of projection for which horizontal range is maximum.

Horizontal range is given by,  

R is maximum if  is maximum.

i.e.      

              

               

Horizontal range is maximum when angle of projection is 45°. 

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There are two angles of projection for which horizontal range is the same. Show that product of their time of flights is directly proportional to the horizontal range?


The angles of projection, for which horizontal range is same, are θ and 90° – θ.

Time of flight for angle of projection θ is,

                       

Time of flight corresponding to angle of projection (90°–θ) is, 

                  

Now, product of their time flight is, 

    

         
        

           
            

That is,

                        

Hence proved the relation. 
                      
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A shot fired horizontally from the top of a tower with a velocity of 60 ft/s hits the ground in 2 seconds. What is the height of the tower and the distance from the foot of the tower where it strikes the ground?

Here,
Time taken for the shot to reach the ground, t = 2s.

Velocity in horizontal direction, u = 60 ft/s

The height of the tower = distance fallen by shot in 2s 

That is, 

 

The distance from the foot of the tower where it strikes the ground, s= ut = 60 x 2 = 120 feet. 

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A ball is dropped from a certain point. At the same time another ball is fired horizontally from the same height. Which one will hit the ground earlier? Explain.

Initial velocity of both the balls in vertical direction is zero. So, height by which both the balls have to fall is the same. Therefore, both the balls will hit the ground simultaneously. 
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Show that there are two angles of projection for which horizontal range is the same.

Horizontal range is given by,

                   
where, straight theta is the angle of projection and is less than 90 degree.

We know, 
             

                     [Where ]

Now,  

           and  

Since     

∴            

Thus horizontal range is the same for angle of projection θ and α = 90°–θ. 

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