Ice skaters, ballet dancers, and basketball players executing vertical leaps often give the illusion of "hanging" almost motionless near the top of the leap. To see why this is, consider a leap to maximum height \(h .\) Of the total time spent in the air, what fraction is spent in the upper half (i.e., at \(y>\frac{1}{2} h\) )?

Projectile motion describes the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.

In the context of a vertical leap, such as that performed by an ice skater or basketball player, the only force acting on the athlete after leaving the ground is gravity. The initial upwards velocity propels them upwards, but gravity slows this ascent until the upward velocity becomes zero at the peak height. Then, the jumper descends back to the ground with increasing speed until they land.

The key point to remember is that projectile motion is two-dimensional motion under constant acceleration due to gravity. This means that vertically, the motion is uniformly accelerated (changing velocity due to gravity), and horizontally, if any, the motion would be at a constant velocity because no horizontal forces are acting on the jumper (assuming air resistance is negligible).

Gravity acceleration, often denoted as 'g', is the acceleration due to gravity near Earth's surface. The standard value of gravity acceleration is approximately 9.81 meters per second squared \( m/s^2 \) downward.

In our scenario, when an athlete performs a vertical leap, they propel themselves against this gravitational pull. Gravity acts to decelerate the athlete's upward velocity uniformly until it becomes zero at the apex of the leap. Upon reaching this peak, gravity then accelerates the athlete back towards the ground.

Significance in Projectile Motion

Gravity plays a pivotal role in projectile motion, as it's the only force acting on the object once it is in motion. Whether the trajectory is vertical or at an angle, gravity acceleration will determine the time it takes to reach the maximum height and the speed upon landing. In vertical leaps, the effect of gravity manifests in a symmetric rise and fall pattern.

The symmetric rise and fall in projectile motion, particularly in a vertical leap, is a fascinating outcome of the constant gravitational acceleration acting on the athlete.

As the athlete jumps upwards, they slow down gradually until they reach the leap's peak, where the velocity is zero. Once they start descending, they gain speed at the same rate they lost it when ascending. Because the force of gravity is consistent and the leap's rise and descent only differ in their direction, the athlete spends the same amount of time ascending to the peak as descending back to Earth.

This symmetry leads to the intuitive result that if the total time in the air is split into four equal parts, two parts will be spent in the upper half of the leap and two in the lower half. Thus, when an athlete seems to 'hang' in the air, they are spending half their total air time above half the leap's maximum height, leading to the 'illusion' of hanging motionless at the top.