"horizontal range" is the component of displacement that lies along a horizontal axis, usually the x-axis.
(i) an object displaces 100 meters along a line that lies at an angle 60º above the +x axis. Then the horizontal range is 100m * cos60º = 50 m.
(ii) a car starts from rest and accelerates "east" at a rate of 4 m/s². After 4 seconds the horizontal range is
½ * 4m/s² * (4s)² = 32 m.
(iii) But most often, "range" is used in "projectile" problems, where an object has been launched at some angle Θ above horizontal with some speed V, and the ground is presumed to be level, so that the impact height is the same as the launch height. Then the horizontal range can be calculated using
R = V²sin(2Θ) / g
For a given launch velocity V, the range R is maximized when Θ = 45º (and sin(2Θ) = 1).
(iv) But sometimes the ground isn't level, and the launch and impact heights differ. To find the range, the easiest route is often to use the trajectory equation and solve for x:
y = h + x·tanΘ - g·x² / (2v²·cos²Θ)
where y = height at x-value of interest
and h = initial height =
and x = range of interest =
and Θ = launch angle =
and v = launch velocity =
(v) and sometimes we simply have objects launched horizontally from some height h (say, from a table top) and you need to know how far it lands from the end of the table. For these you need to find the time it takes to fall:
h = ½gt² → t = √(2h / g)
and then plug the time into
x = v*t
where v was the (horizontal) velocity at launch.
That's all the basic examples I can think of.