Problem 25 - University Physics with Modern Physics

Question

In the movie Superman, Lois Lane falls from a building and is caught by the diving superhero. Assuming that Lois, with a mass of \(50.0 \mathrm{~kg}\), is falling at a terminal velocity of \(60.0 \mathrm{~m} / \mathrm{s}\), how much force does Superman exert on her if it takes \(0.100 \mathrm{~s}\) to slow her to a stop? If Lois can withstand a maximum acceleration of \(7 g^{\prime}\) s, what minimum time should it take Superman to stop her after he begins to slow her down?

Step-by-Step Solution

Answer

Answer: Superman exerts a force of 30,000 N on Lois Lane to bring her to a stop, and the minimum time required to stop her without exceeding her maximum acceleration is 0.874 seconds.

Step 1: Calculate the initial force exerted on Lois

To determine the force exerted on Lois, we'll use Newton's second law of motion, which states that the force F is equal to mass m times acceleration a. In this situation, her mass is \(50.0kg\) and the acceleration is obtained by calculating the change in velocity (from her initial terminal velocity to a full stop) divided by the time it takes to slow her down. The initial terminal velocity is given as \(60.0m/s\), and the time is \(0.100s\). The formula for calculating force is: \(F = ma\) where \(F\) = force exerted on Lois \(m\) = mass of Lois \(a\) = acceleration We need to find acceleration first. We can use the formula: \(a = \frac{v_f - v_i}{t}\) where \(a\) = acceleration \(v_f\) = final velocity (0m/s, since she is stopped) \(v_i\) = initial velocity (60.0m/s, given as her terminal velocity) \(t\) = time (0.100s, given)

Step 2: Calculate the acceleration

Now, we'll calculate the acceleration by substituting the given values: \(a = \frac{0 - 60.0}{0.100} = \frac{-60.0}{0.100}= -600.0 \mathrm{~m}/\mathrm{s}^2 \) Now that we have the acceleration, we can find the force exerted on Lois.

Step 3: Calculate the force exerted on Lois

Using the force formula, we'll substitute the values for mass and acceleration: \(F = (50.0 \mathrm{~kg})(-600.0 \mathrm{~m}/\mathrm{s}^2) = -30000 \mathrm{~N}\) The force Superman exerts on Lois to bring her to a stop is \(30000 N\) in the upward direction.

Step 4: Calculate the minimum time needed

Now, we will find the minimum time needed to stop Lois without exceeding her maximum acceleration. Lois' maximum acceleration can withstand is given as \(7g'\), where g is the gravitational acceleration, approximately \(9.81 \mathrm{~m}/\mathrm{s}^2\). Thus, her maximum acceleration is: \(max\_a = 7 \times 9.81 = 68.67 \mathrm{~m}/\mathrm{s}^2\) Now, we have to find the minimum time necessary. We can rearrange the acceleration formula to find the time: \(t = \frac{v_f - v_i}{a}\) where \(t\) = minimum time to stop Lois \(v_f\) = final velocity (0m/s, since she will be stopped) \(v_i\) = initial velocity (60.0m/s, given as her terminal velocity) \(a\) = maximum acceleration (68.67m/s^2, as calculated above)

Step 5: Calculate the minimum time

Now we substitute the values into the formula to find the minimum time: \(t = \frac{0 - 60}{68.67} = \frac{-60.0}{68.67} ≈ -0.874 \mathrm{~s}\) Since time should be positive, we will take the absolute value: \(t = 0.874 \mathrm{~s}\) Therefore, Superman should take at least \(0.874\) seconds to stop Lois after he starts slowing her down to ensure her maximum acceleration isn't exceeded.

Key Concepts

Acceleration

When discussing motion, acceleration plays a critical role. It tells us how quickly an object changes its velocity. In the scenario with Lois Lane, acceleration occurs as Superman decelerates her from a high-speed fall to a complete stop. This change is vital to analyze. The basic formula used for calculating acceleration is: \[ a = \frac{v_f - v_i}{t} \] Where:

• \(a\) = acceleration
• \(v_f\) = final velocity (for Lois, it is 0 m/s, as she stops)
• \(v_i\) = initial velocity (in this case, 60.0 m/s)
• \(t\) = time period over which this change occurs (0.100 seconds in this problem)

By substituting in the known values, we see that Lois experiences a dramatic change in velocity over a very short time frame. This scenario highlights how acceleration isn't just about speeding up but also slowing down.

Force Calculation

Newton's Second Law of Motion provides a clear path to understand forces involved in motion. This law is captured using the equation: \[ F = ma \] In the Lois Lane example, it's necessary to calculate the force Superman must exert to safely stop her fall. Here's what's involved: - **Mass** - Lois's mass impacts how much force is needed. In this case, her mass is 50.0 kg.- **Acceleration** - Previously calculated as \(-600.0 \mathrm{~m/s}^2\) from her initial terminal velocity to a stop.By multiplying mass and acceleration, we derive the force. Substituting the values gives: \[ F = (50.0 \mathrm{~kg})(-600.0 \mathrm{~m/s}^2) = -30000 \mathrm{~N} \] The negative sign indicates the direction is opposite to Lois's fall, meaning the force is upwards. This immense force underscores the strength Superman needs.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. For Lois Lane, as she falls, air resistance reaches a point where it balances the force of gravity, leading her to fall at a steady terminal velocity of 60.0 m/s. Understanding terminal velocity involves noting a few critical factors:

• **Mass of the object**: Heavier objects have a higher terminal velocity given more force is required to balance gravity.
• **Air resistance**: This force opposes motion and depends on shape, size, and surface, explaining why falling objects reach a steady speed.

In Lois's case, knowing her terminal velocity helps Superman calculate the right amount of force and the required time to decelerate her safely when catching her. This showcases how physics principles guide heroic acts even in movies.