Range of a Projectile

The next question we can try to answer is what is the range of a projectile or how far will the object go?  We can find this by first figuring out how long it will be or was in the air.  We start by observing that a projectile will be at y equal to zero at two times.  Once when it first leaves the ground and once at the point when it lands.  The first situation will be at t equal to zero and the second situation will be when t equals the time at spent in the air.  So setting y equal to zero in our equation for vertical position:

We then substitute this value of t into our position in x equation and do some algebra:

combine the vo terms

apply the following trigonometric identity

The final equation is that for the range of the projectile.  Notice that if initial velocity is held constant the range will only vary with the angle.  So this range will be a maximum when the sine of two theta is a maximum.  Since sine goes between one and zero we want the sine of two theta to equal one and this occurs when theta is 45 degrees.  So one can achieve maximum range by shooting a projectile at 45 degrees.  Other angles above or below 45 degrees will result in lesser ranges.  Also note that because of this fact, I can obtain the same range with two different angles:

Same Range from two Different Angles

Peak Height of a Projectile
 
 

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