Problem 71 - Calculus and its applications

Question

On Earth, an object travels \(4.905 \mathrm{~m}\) after 1 sec of free fall. Thus, by symmetry, an athlete would require 1 sec to jump \(4.905 \mathrm{~m}\) high, and another second to come back down. Is it possible for a person to stay in the air for (have a "hang time" of) 2 sec? Can a person have a hang time of 1.5 sec? 1 sec? What do you think is the longest possible hang time achievable by humans jumping from level ground?

Step-by-Step Solution

Answer

Yes to 1 sec hang time; 2 sec is unrealistic for humans; longest realistic is around 1 sec.

Step 1: Understanding Free Fall and Hang Time

Free fall is the motion of any object under the influence of gravity only, with no other forces acting on it. For an object in free fall, the distance it travels in time can be calculated using the equation: \( h = \frac{1}{2} g t^2 \) where \( h \) is the height traveled, \( g \) is the acceleration due to gravity \( (9.81 \ \text{m/s}^2) \), and \( t \) is the time.

Step 2: Calculating Maximum Height and Hang Time

If an athlete can jump to a height of \( 4.905 \ \text{m} \), the time to reach that height is found using \( h = \frac{1}{2} g t^2 \). Solving for \( t \) gives \( t = \sqrt{\frac{2h}{g}} \). Substituting \( h = 4.905 \) m and \( g = 9.81 \ \text{m/s}^2 \), we find \( t = 1 \) second to reach maximum height. Since time to ascend equals time to descend, total hang time is \( 2 \) seconds for this height.

Step 3: Evaluating Different Hang Times

A hang time of \( 2 \) seconds is possible as shown, since \( 1 \) second is taken to ascend and \( 1 \) second to descend. For a hang time of \( 1.5 \) seconds, each phase (ascend or descend) would be \( 0.75 \) seconds, implying a reach to smaller height than \( 4.905 \) m. For hang time of \( 1 \) second, each phase would be \( 0.5 \) seconds, reaching an even lesser height, which is more plausible. Physically, only the \( 1 \) second hang time or less is realistic for level ground jumping due to typical human capabilities.

Step 4: Checking Maximum Hang Time

The hang time of \( 2 \) seconds corresponds to an exceptional athletic jump to a \( 4.905 \ \text{m} \) height, far greater than typical human performance, even for elite athletes. Therefore, the longest possible hang time achievable by humans, considering physical limits and jumping from level ground, is significantly less than \( 2 \) seconds, likely closer to \( 1 \) second.

Key Concepts

Free Fall

Free fall is a fascinating concept in physics that describes the motion of an object solely under the influence of gravity, without interference from other forces like air resistance. When an object is in free fall, it accelerates downwards at a constant rate due to gravitational pull.

To calculate the distance traveled by an object in free fall over a period of time, we use the formula:

• \( h = \frac{1}{2} g t^2 \)

Here, \( h \) represents the height or distance traveled, \( g \) is the acceleration due to gravity, which is approximately \( 9.81 \ ext{m/s}^2 \) on Earth, and \( t \) is the time in seconds.

This simple equation helps us understand how objects move when only gravity is acting upon them, providing insights into a range of phenomena from falling apples to the moon’s orbit.

Acceleration Due to Gravity

The acceleration due to gravity is a critical factor in the study of motion, affecting everything from everyday activities to scientific analyses. On Earth, this acceleration is typically \( 9.81 \ ext{m/s}^2 \), meaning that an object's speed increases by 9.81 meters per second every second it is in free fall.

This constant acceleration plays a vital role in determining how quickly an object will fall and the distance it will cover over time:

• Equations of motion often rely on a known value of gravitational acceleration, simplifying complex calculations.
• Understanding this acceleration is key to predicting how objects behave in free fall and their subsequent motion.

Whether jumping from a ledge, calculating projectile paths, or designing spacecraft, gravity remains a fundamental force influencing everything we do.

Human Jumping Capabilities

Human jumping capabilities present an interesting intersection of physics and biology. While humans are capable of impressive feats, there are natural limits dictated by physics and our own physiology that constrain jumping height and hang time.

When a person jumps, they exert force against the ground, propelling their body upward. The maximum height achieved during a jump is determined by the force of the jump, the athlete's strength, and gravitational pull:

• The typical human jump results in hang time of less than a second.
• Elite athletes can maximize their hang time through training, potentially achieving a hang time of up to 1 second from level ground, though not significantly more.

The idea of a 2-second hang time, as discussed in theoretical physics examples, often exceeds realistic limits for humans. Training and technology may enhance performance to a degree, but gravitational forces present a natural ceiling for human jumping achievements.