Problem 42 - Calculus

Question

(a) If you jump out of an airplane without a parachute, you fall faster and faster until air resistance causes you to approach a steady velocity, called the terminal velocity. Sketch a graph of your velocity against time. (b) Explain the concavity of your graph. (c) Assuming air resistance to be negligible at \(t=0\) what natural phenomenon is represented by the slope of the graph at \(t=0 ?\)

Step-by-Step Solution

Answer

(a) Graph is initially concave down, flattening to horizontal. (b) Concave down shows decreasing acceleration. (c) Initial slope is gravity, \( g \).

Step 1: Understanding the Scenario

When you jump out of an airplane, gravity causes you to accelerate downwards. Over time, air resistance increases until it balances the gravitational force, leading to a constant terminal velocity. We need to graph velocity on the y-axis and time on the x-axis.

Step 2: Sketching the Velocity-Time Graph

Start at the origin of the graph, where time (t=0) and initial velocity (v=0). As time progresses, the velocity increases rapidly due to gravity. As air resistance grows, the rate of increase in velocity diminishes, forming a curve that flattens out at the terminal velocity, which means the velocity approaches a horizontal asymptote as time increases.

Step 3: Explaining the Concavity of the Graph

Initially, the graph is concave downwards. This reflects the decrease in acceleration as air resistance increases. In mathematical terms, the second derivative of velocity with respect to time is negative, indicating that the rate at which velocity increases (acceleration) is decreasing.

Step 4: Analyzing the Initial Slope

At time t=0, air resistance is negligible, so the only force acting is gravity. The slope of the velocity-time graph at t=0 represents acceleration due to gravity, a constant value denoted as \( g \approx 9.8 \text{m/s}^2 \). This shows how rapidly velocity is increasing at the start of the fall.

Key Concepts

Air Resistance

Air resistance is a crucial factor in understanding how objects move through the air, especially when considering velocity-time graphs. When you drop an object, like a person jumping out of an airplane, the air pushes against it. This push is what we call air resistance. It acts in the opposite direction of the motion, trying to slow you down.

In the context of free fall, air resistance starts small but grows as you speed up. Why does it increase? Because the faster you move, the more air you "hit" per second, and all those air molecules push back harder. So, air resistance just isn't static; it's dynamic and varies with speed.

Air resistance aims to counteract the force pulling you down, which is gravity. Initially, gravity wins and you accelerate downwards. But, as air resistance grows, it gradually reduces this downward pull's effect.

• This change in forces is why velocity-time graphs for falling objects curve.
• The graph starts steep and slowly becomes less steep as resistance builds.
• The bending graph signifies the growing influence of air resistance until it matches gravity.

When this balancing happens, you reach what's called terminal velocity.

Terminal Velocity

Imagine falling from a great height without getting infinitely faster; how does that work? Terminal velocity is the answer. It's the constant speed you reach when the force of air resistance equals the gravitational pull.

In practical terms, terminal velocity is that final speed you hit when you can't speed up any more during a fall. Once air resistance cancels out the acceleration due to gravity, there's no longer an unbalanced force accelerating you.

At this point, your velocity-time graph levels out. The curve stops rising and becomes a flat line, showcasing that your speed is steady.

• The terminal velocity depends on several factors such as your body's shape, mass, and the air density.
• A flatbody will encounter more resistance, often reducing terminal speed.
• Heavier bodies, on the other hand, might reach higher terminal velocities due to the increased gravitational force.

Terminal velocity ensures you fall safely, though fast; giving you that steady rate much needed for calculations and predictions in physics.

Acceleration Due to Gravity

When talking about objects in free fall, one of the fundamental terms is acceleration due to gravity. This is the rate at which an object speeds up when it falls freely, without the interference of other forces like air resistance at first.

Known by the symbol \( g \), its value on Earth averages around \( 9.8 \text{ m/s}^2 \). This means that every second, your speed increases by about 9.8 meters per second when only gravity influences your motion.

In a velocity-time graph, this rate is represented as the initial steepness of the curve. The graph starts with a sharp upward slope, indicating rapid acceleration. Over time, as air resistance kicks in, that steepness reduces.

• At \( t=0 \), the graph displays the maximum slope.
• This initial slope reflects the pure, unopposed gravitational pull.
• As seconds pass, the graph's slope gently declines as air resistance competes with gravity.

Understanding \( g \) is essential for comprehending how objects fall and come to stop accelerating once forces balance out.