A particle is projected with speed ‘u’ at an angle θ. Show that it will strike the ground with the speed u.
            

Let a projectile be projected with speed 'u' at angle 'θ' from O and strike the ground at P with velocity 'v'.

The projectile will strike the ground at P after time equal to the time of the flight.

Now, resolving the components of 'u' in the horizontal and vertical direction, we have

                   

Horizontal and vertical component is given by, 

          

Therefore magnitude of velocity at ground is, 

         

                    

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A particle is projected with velocity ‘u’ in two different ways: first such that horizontal range is maximum and second such that the height is maximum. What is the ratio of maximum height to the maximum horizontal range?

Horizontal range is maximum for angle of projection 45°.

Maximum horizontal range is,

              

The height is maximum, if the projectile is projected in vertically upward direction, i.e. θ = 90°. 

Maximum height is, 

          

So, the ratio of maximum height to the maximum horizontal range is given by, 

        
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A projectile is projected with velocity u at an angle θ with the vertical. At what point velocity and acceleration will be perpendicular to each other?

At the highest point the velocity is in horizontal direction and acceleration is in vertically downward direction. Hence, velocity and acceleration of projectile are perpendicular to each other at the highest point. 
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In long jumping does it matter how high you jump? What factor determines the span of the jump?

It does not matter how high an athlete jumps in a long jump. The span of long jump depends upon the initial velocity and the angle at which an athlete jumps. The span of jump is maximum for an angle of 45°.
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Why one has to aim a little above the target?


If one directly aims at the target as shown in figure (i), the bullet will miss the target because due to gravity it will follow parabolic path.

If one directly aims at the target as shown in figure (i), the bullet

Therefore to hit the target one has to aim a little above as shown in figure (ii), so that after following the parabolic path, the bullet may hit the target.

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