Question

A pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of 143 at an angle of 47.8 above the horizontal. You can ignore air resistance. Find the distance from its firing point to its landing point if it is fired on the level salt flats of Utah.

Answer 1

We know that the horizontal velocity doesn't change because no force acts in that direction. Therefore, we get the following expression for horizontal displacement: \[x = v t \cos(\theta)\]

The expression for vertical displacement is: \[y = v t \sin(\theta) - {1 \over 2} g t ^2\]

We need to find the time it takes for the projectile to fly through the air. This is accomplished by finding the time at which y=0. This will be\[0 = v t \sin(\theta) - {1 \over 2} g t^2 \rightarrow t = {2 v \sin(\theta) \over g} \]

Now, we can find how far the projectile traveled in the horizontal by plugging in the above expression for time into the horizontal displacement equation. \[x = {2 v^2 \sin(\theta) \cos(\theta) \over g}\]Note that \(2 \sin(\theta) \cos(\theta) = sin(2 \theta)\)\[x = {v^2 \sin(2 \theta) \over g}\]