Two directions, one flight.

Resolve a launch velocity into horizontal u $\cos\theta$ and vertical u $\sin\theta$ components. Horizontal velocity remains constant when no horizontal force acts; vertical velocity changes under gravity. Both components share the same elapsed time.

Choose a vertical sign convention before applying the constant-acceleration equations. At maximum height vertical velocity is zero, while horizontal velocity remains unchanged. The familiar range expression assumes equal launch and landing heights; it cannot be used unmodified for a cliff launch.

INTERACTIVE MODEL

Launch and landing

Range: 40.82 m · Flight time: 2.89 sLaunch angle / degrees45

READ THE GRAPH

Explore the relationship

Launch speed 20 m s⁻¹, same launch and landing height, g = 9.8 m s⁻², no air resistance.

Height / m

Tap or focus a plotted point to read its values.

View plotted data as a table

Explore the relationship
Series	Fraction of flight time	Height / m
Height / m	0	0
Height / m	0.0166667	0.668934
Height / m	0.0333333	1.31519
Height / m	0.05	1.93878
Height / m	0.0666667	2.53968
Height / m	0.0833333	3.11791
Height / m	0.1	3.67347
Height / m	0.116667	4.20635
Height / m	0.133333	4.71655
Height / m	0.15	5.20408
Height / m	0.166667	5.66893
Height / m	0.183333	6.11111
Height / m	0.2	6.53061
Height / m	0.216667	6.92744
Height / m	0.233333	7.30159
Height / m	0.25	7.65306
Height / m	0.266667	7.98186
Height / m	0.283333	8.28798
Height / m	0.3	8.57143
Height / m	0.316667	8.8322
Height / m	0.333333	9.07029
Height / m	0.35	9.28571
Height / m	0.366667	9.47846
Height / m	0.383333	9.64853
Height / m	0.4	9.79592
Height / m	0.416667	9.92063
Height / m	0.433333	10.0227
Height / m	0.45	10.102
Height / m	0.466667	10.1587
Height / m	0.483333	10.1927
Height / m	0.5	10.2041
Height / m	0.516667	10.1927
Height / m	0.533333	10.1587
Height / m	0.55	10.102
Height / m	0.566667	10.0227
Height / m	0.583333	9.92063
Height / m	0.6	9.79592
Height / m	0.616667	9.64853
Height / m	0.633333	9.47846
Height / m	0.65	9.28571
Height / m	0.666667	9.07029
Height / m	0.683333	8.8322
Height / m	0.7	8.57143
Height / m	0.716667	8.28798
Height / m	0.733333	7.98186
Height / m	0.75	7.65306
Height / m	0.766667	7.30159
Height / m	0.783333	6.92744
Height / m	0.8	6.53061
Height / m	0.816667	6.11111
Height / m	0.833333	5.66893
Height / m	0.85	5.20408
Height / m	0.866667	4.71655
Height / m	0.883333	4.20635
Height / m	0.9	3.67347
Height / m	0.916667	3.11791
Height / m	0.933333	2.53968
Height / m	0.95	1.93878
Height / m	0.966667	1.31519
Height / m	0.983333	0.668934
Height / m	1	0

Probe along the horizontal axis0Height / m: 0

Explore: Compare complementary launch angles. What stays equal about the range?

u_x=u\cos\theta\quad u_y=u\sin\theta\quad x=u_xt\quad y=u_yt-\tfrac12gt^2

WORKED EXAMPLE

Launch horizontally at 6 $\mathrm{m}\,\mathrm{s}^{-1}$ from 19.6 m above the ground; g = 9.8 $\mathrm{m}\,\mathrm{s}^{-2}$ .

19.6 = 0.5 × 9.8 × $t^{2}$ gives t = 2 s.
Horizontal range = 6 × 2 = 12 m.

Assumed knowledge

Trigonometry, vectors and constant acceleration.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Resolve and recombine perpendicular vectors.
Solve projectile motion without drag.

QCAA Physics · Unit 3 · Gravity and motion ↗

READY TO TRY IT?

Two directions, one flight.

Launch and landing

Explore the relationship

Launch horizontally at 6 m s−1\mathrm{m}\,\mathrm{s}^{-1}ms−1 from 19.6 m above the ground; g = 9.8 m s−2\mathrm{m}\,\mathrm{s}^{-2}ms−2.

Put the idea to work.

Launch horizontally at 6 $\mathrm{m}\,\mathrm{s}^{-1}$ from 19.6 m above the ground; g = 9.8 $\mathrm{m}\,\mathrm{s}^{-2}$ .