Given a projectile is launched, from rest, at an angle θ and travels at a velocity V, what is the range and path of motion of the projectile? (Ignore air resistance.)

First, find the formula for the time taken, t, for the projectile to travel the distance. Using the fact that the projectile reaches a velocity of zero at a time of 0.5t when at its maximum height and acceleration due to gravity is negative, the time of flight is dependent on vertical values so; v=u+at => 0=Vsinθ-g(0.5t) => t=(2Vsinθ)/g. Now for the range, also know as maximum displacement, substitute the time taken into the distance, x, formula with horizontal values; x=ut => x=Vcosθt => x(max)=Vcosθ((2Vsinθ)/g)) => x(max)=Range=(2sinθcosθ(V)^2)/g. Using trigonometric identity sin2θ=2sinθcosθ, we have Range=(sin2θ(V)^2)/g. To find the motion of the projectile, use the equation for displacement s=ut+0.5a(t)^2, therefore in vertical terms y=Vsinθ(t)-0.5g(t)^2, and thus insert the horizontal time taken which is derived from x=Vcosθt => t=x/Vcosθ, so the path the projectile follows on the x-y plane is y=Vsinθ(x/Vcosθ)-(0.5g(x)^2)/(Vcosθ)^2. Tidying this up, and using the fact that secθ=1/cosθ and tanθ=sinθ/cosθ; this means y=xtanθ-g((xsecθ)^2)/2(V)^2. Since the equation is in the form y=ax-bx^2, for some a,b, the motion of the projectile must be parabolic. And we are done.

OD
Answered by Oskar D. Physics tutor

5929 Views

See similar Physics A Level tutors

Related Physics A Level answers

All answers ▸

How would you prove the formula for the total capacitance of a system consisting of several capacitors linked in series?


Define a geostationary orbit


Two cars start at point A. Car 1 moves in a direction at 5 m/s. After 10 seconds car 2 accelerates in the same direction as car 1 at 2m/s^2. At what time after car 1 starts moving and distance from A does car 2 pass car 1?


Please see below.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning