Projectile Landing Game

About Landing

Landing shows the first half of a throw: a ball leaves a point on the left and climbs in a curve, which stops at the highest point of its flight. The ground is marked with distances in meters. You slide a red marker along the ground to where you think the ball will land. There is no wind and no air resistance.

On flat ground the answer is a neat trick of symmetry, because the second half of the flight mirrors the first, so the landing spot is twice as far as the top of the arc. From level 5 the throw sometimes starts from a cliff between 4 and 30 meters high, and then the ball falls further than it climbed and the simple doubling no longer works.

How to play

1. Look at the arc and find its highest point. The distance from the launch point to the top is your first clue.
2. Decide whether the throw starts at ground level or from a cliff, which is stated under the question.
3. Slide the marker to the spot where you think the ball will touch the ground. The distance scale along the ground helps with the numbers.
4. Lock in. The full flight path is drawn so that you can see how it ended.

You see a ball's flight up to its highest point. Slide the red marker to where it will hit the ground. No air resistance.

How scoring works

The score depends on how far your marker is from the true landing point, relative to the length of the throw. You get 100 for a perfect answer and 0 when you are 42 percent of the range away on level 1, or 18 percent on level 10. So on a 60 meter throw at the top level, an error of 5 meters still earns about 54 points.

The launch speed is between 11 and 32 meters per second and the angle between 18 and 62 degrees, which leads to ranges between roughly 10 and 110 meters. You have 18 seconds on the first level and 14 on the last.

What it trains

This game trains the intuition behind catching a ball, throwing at a target, kicking a free kick or judging where a thrown object will land. Our brains do this with surprising skill in real time, but explaining it, or doing it with unusual conditions such as a cliff, is much harder.

It also trains the use of symmetry and extrapolation, two skills that are useful far beyond ballistics: seeing the second half of a pattern from the first half.

Tips for a higher score

• On flat ground, double the distance to the top. The ball spends the same time and covers the same horizontal distance going down as it did going up.
• From a cliff, add the extra time. The ball keeps moving sideways while it falls the additional height, so the landing point is further than the doubled distance.
• Use the fall time rule: a drop of 5 meters takes about 1 second, 20 meters about 2 seconds and 45 meters about 3 seconds. Multiply by the sideways speed.
• Get the sideways speed from the arc: the horizontal distance to the top divided by the time to climb gives how fast the ball moves sideways.
• Do not overshoot out of caution. Players tend to place the marker too far when the arc looks flat and too close when it looks steep; check which way you lean.

The idea behind it

Ignoring air resistance, a projectile moves with constant speed horizontally and with constant downward acceleration of 9.81 meters per second squared vertically. Galileo described this in his 1638 book Two New Sciences and showed that the path is a parabola. The two motions can be treated separately.

On level ground the maximum range is reached with a launch angle of 45 degrees, and angles that add to 90 degrees, such as 30 and 60, give the same range. In real life, drag shortens the flight of fast or light objects, which is why the game talks about an ideal throw.

Related games

Free, no account. Your best scores are saved on your device.