Projectile motion and kinematics (Class 11): graphs, vectors and trajectories
Kinematics describes motion without asking what causes it. A projectile moves with constant velocity along x and constant acceleration g along y. Change the angle and speed and see range, height and time of flight change.
NCERT Class 11 Physics, Chapters 2 and 3 (Motion in a Straight Line, Motion in a Plane). Every lab below runs in 3D in your browser or on a phone, in English and Hindi.
Try the labs
Projectile motion with optional air drag · plan · Chapter 3
To study motion of a projectile as the superposition of uniform motion along x and uniformly accelerated motion along y, and to verify the formulas for time of flight, maximum height and range.
x-t, v-t, a-t graphs of a moving car · free · Chapter 2
To understand motion graphs: relation between position, velocity and acceleration through slopes and areas.
River crossing: shortest time vs shortest path · plan · Chapter 3
To study motion of a boat in a river using relative velocity: minimum time of crossing and shortest path.
Relative velocity: two trains and a river boat · plan · Chapter 2
To understand relative velocity and frames of reference by studying two moving trains.
Free fall with and without air drag · plan · Chapter 2
To study motion under gravity (free fall), verify the equations of uniformly accelerated motion and see how air resistance changes the motion.
Projectile on an inclined plane · plan · Chapter 3
To analyse projectile motion on an inclined plane: range along the slope, time of flight, maximum height from the slope and the angle for maximum range.
Uniform circular motion and centripetal acceleration · plan · Chapter 3
To study uniform circular motion, understand centripetal acceleration and its dependence on speed and radius.
Vector addition: triangle and parallelogram law · free · Chapter 3
To add and subtract two vectors graphically and analytically, using the triangle and parallelogram laws, and to resolve a vector into components.
Equations of motion
For constant acceleration a: v = u + at, s = ut + ½at² and v² = u² + 2as. The slope of the x–t graph is the velocity and the slope of the v–t graph is the acceleration; the area under the v–t graph is the displacement.
v = u + ats = ut + ½ a t²v² = u² + 2as
Projectile motion
A body thrown at speed u at angle θ has horizontal velocity u cos θ, unchanged, and vertical velocity u sin θ − gt. The range is greatest at 45° on level ground, and angles θ and 90° − θ give the same range.
T = 2u sin θ / gH = u² sin²θ / 2gR = u² sin 2θ / gy = x tan θ − g x² / (2u² cos²θ)
Relative velocity and rivers
The velocity of A relative to B is vA − vB as vectors. A swimmer who heads straight across a river is carried downstream; to reach the point directly opposite he must aim upstream at sin θ = vr/vb. The shortest time is always for the straight heading.
v_AB = v_A − v_Bt = d / vb (shortest time)sin θ = vr / vb (shortest path)
Frequently asked questions
Which angle gives the maximum range?
45° on level ground when air resistance is ignored. On an incline or with a different landing height the angle changes.
Is the speed of a projectile minimum at the top?
Yes. The vertical velocity is zero there, so the speed is u cos θ, the smallest value.
What is the path of a projectile?
A parabola, because x is proportional to t and y to t².
Why do two stones dropped from different heights fall with the same acceleration?
Both have the acceleration g in the absence of air resistance; mass does not matter.