Question

In skydiving, the vertical velocity component of the skydiver is typically zero at the moment he or she leaves the plane; the vertical component of the velocity then increases until the skydiver reaches terminal speed (see Chapter 4). For a simplified model of this motion, we assume that the horizontal velocity component is zero and that the vertical velocity component increases linearly, with acceleration \(a_{y}=-g,\) until the skydiver reaches terminal velocity, after which it stays constant. Thus, our simplified model assumes free fall without air resistance followed by falling at constant speed. Sketch the kinetic energy, potential energy, and total energy as a function of time for this model.

Short answer

Short Answer: During the first phase of free fall, kinetic energy (KE) increases quadratically, while potential energy (PE) decreases quadratically. Total energy (TE) remains constant. During the second phase, when the skydiver reaches terminal velocity and continues at a constant speed, kinetic energy (KE) remains constant, potential energy (PE) decreases linearly, and total energy (TE) also decreases linearly. To sketch the graphs, use the expressions derived for KE, PE, and TE in both phases and show the change in energy as a function of time.

Step-by-step solution

Phase 1: Free Fall

During free fall, the skydiver accelerates downward with \(a_y = -g\), where \(g\) is the acceleration due to gravity. To find expressions for the kinetic, potential, and total energies during free fall, we first need to find the vertical velocity (\(v_y\)) and displacement (\(h\)) as a function of time. Using the kinematic equation \(v_y = u + at\), where \(u\) is the initial velocity, which is 0. So \(v_y = -gt\). Now, for displacement, we use the equation \(h = ut + \frac{1}{2}at^2\), since \(u = 0\). So, \(h = -\frac{1}{2}gt^2\). Now we can find the energies. 1. Kinetic Energy (KE): \(KE = \frac{1}{2}mv^2 = \frac{1}{2}m(-gt)^2 = \frac{1}{2}mgt^2\) 2. Potential Energy (PE): \(PE = mgh = mg(-\frac{1}{2}gt^2) = -\frac{1}{2}mg^2t^2\) 3. Total Energy (TE): \(TE = KE + PE = \frac{1}{2}mgt^2 - \frac{1}{2}mg^2t^2\)

Phase 2: Constant Speed

Once the skydiver reaches terminal velocity (\(v_t\)), the vertical velocity component remains constant. Let's assume the skydiver reaches terminal velocity after time \(t_t\), and the height at that moment is \(h_t\). For the constant speed phase, the displacement can be given by \(h = h_t - v_t(t - t_t)\). Now we can find the energies. 1. Kinetic Energy (KE): \(KE = \frac{1}{2}mv^2 = \frac{1}{2}mv_t^2\) (constant) 2. Potential Energy (PE): \(PE = mgh = mg(h_t - v_t(t-t_t)) = mgh_t - mgv_t(t - t_t)\) 3. Total Energy (TE): \(TE = KE + PE = \frac{1}{2}mv_t^2 + mgh_t - mgv_t(t - t_t)\)

Sketching the Graphs

Now we are ready to sketch the graphs of KE, PE, and TE as a function of time based on the expressions derived for both phases. 1. Kinetic Energy (KE): During free fall, KE increases quadratically until terminal velocity is reached. Then, KE remains constant as the vertical velocity stays constant. 2. Potential Energy (PE): During free fall, PE decreases quadratically but at a higher rate than KE. Once terminal velocity is reached, the PE decreases linearly as height decreases linearly with time. 3. Total Energy (TE): During free fall, the total energy of the skydiver remains constant as the difference between the quadratic terms for KE and PE and equal magnitude. Once terminal velocity is reached, the total energy decreases linearly as a result of the linear decrease in PE. Please sketch these graphs using their respective expressions for both the phases.

Key concepts

Free Fall Acceleration

Imagine jumping out of a plane with a parachute on your back, feeling the rush of the air, and seeing the world from a bird's eye view. This is the experience of free fall, a core concept in skydiving physics. But what exactly happens during free fall?

Free fall acceleration refers to the acceleration of an object solely due to the force of gravity. For skydivers, this is experienced when they first jump from the airplane and before they reach terminal velocity. On Earth, the free fall acceleration is approximately \( g = 9.8\, m/s^2 \) downward, providing the skydiver accelerates until they are stopped by air resistance or by opening their parachute.

To simplify the analysis, skydiving physicists often assume no air resistance during the initial phase of the jump—giving us a pure free fall scenario. The skydiver's velocity during this stage increases at a rate of \( g \) for as long as they are in free fall, which means their speed, and subsequently kinetic energy, increases over time.

Kinetic Energy

Kinetic energy refers to the energy that an object possesses due to its motion. For a skydiving physicist, this tends to be one of the most thrilling aspects to explain because it's literally energy in action!

During the free fall phase of a skydive, as the skydiver accelerates towards the Earth, their kinetic energy increases. This increase continues until they reach terminal velocity. The equation that captures this is \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the skydiver. Initially, when the skydiver leaves the plane, \( v = 0 \) and thus \( KE = 0 \) as well. But as they fall and their speed increases under the pull of gravity, their kinetic energy ramps up rapidly, following the equation \( KE = \frac{1}{2}mg^2t^2 \).

Understanding this Energy Transition

Where does this increasing kinetic energy come from? It's converted from potential energy, which decreases as the skydiver falls and loses altitude. This transformation from potential to kinetic energy follows the principles of energy conservation, a fundamental law in physics.

Potential Energy

Potential energy in the context of skydiving is the stored energy that a skydiver has due to their position high above the Earth. It's like having a bank of energy that gets 'spent' as they fall back down towards the surface.

In a skydive, a potential energy is calculated using the equation \( PE = mgh \), where \( m \) is the mass of the skydiver, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground. As the skydiver descends, the height \( h \) decreases, thus reducing their potential energy. During the initial free fall, potential energy converts into kinetic energy, reflecting a shift from stored energy to the energy of motion.

Energy Conservation

Despite these changes, the total energy remains constant (as long as no external work is done on the system). This shows that as potential energy decreases, kinetic energy must increase by the same amount if we ignore air resistance in our initial calculations.

Terminal Velocity

Terminal velocity is the maximum velocity that a falling object can reach when the forces of gravity and air resistance are equal and opposite. It’s the point where the thrill of acceleration ceases, and you are no longer speeding up as you descend through the sky.

At terminal velocity, the skydiver's acceleration due to gravity is completely balanced by the air resistance they experience, resulting in a constant velocity. For an average person in the belly-to-earth position, this speed is around \( 120\, mph \) (or \( 53\, m/s \) ). From a physics standpoint, once the skydiver hits terminal velocity, their kinetic energy no longer increases because their speed stays the same. Subsequently, their potential energy continues to decrease at a uniform rate, given they are still falling towards earth and losing altitude.

As per the step-by-step solution provided, the kinetic energy after reaching terminal velocity remains constant, reflecting the constant speed during this phase. Thus, to a skydiving physicist, terminal velocity signifies not just the limit of speed but also the steadiness of kinetic energy until the parachute is deployed or the free fall ends.