Problem 90 - University Physics with Modern Physics

Question

You are part of a design team for future exploration of the planet Mars, where \(g=3.7 \mathrm{m} / \mathrm{s}^{2} .\) An explorer is to step out of a survey vehicle traveling horizontally at 33 \(\mathrm{m} / \mathrm{s}\) when it is 1200 \(\mathrm{m}\) above the surface and then freely for 20 \(\mathrm{s}\) . At that time, a portable advanced propulsion system (PAPS) is to exert a constant force that will decrease the explorer's speed to zero at the instant she touches the surface. The total mass (explorer, suit, equipment, and PAPS) is 150 \(\mathrm{kg} .\) Assume the change in mass of the PAPS to be negligible. Find the horizontal and vertical components of the force the PAPS must exert, and for what interval of time the PAPS must exert it. You can ignore air resistance.

Step-by-Step Solution

Answer

Vertical force: 894 N; Horizontal force: 399 N; PAPS time: 12.42 s after 20 s free fall.

Step 1: Determine Final Vertical Speed Before Using PAPS

First, calculate the vertical speed of the explorer just before the PAPS engages. The initial vertical speed is 0 since the explorer is stepping out of a horizontally moving vehicle.Use the formula for final velocity: \[ v = u + gt \]Where \( u = 0 \), \( g = 3.7 \, \mathrm{m/s^2} \), and \( t = 20 \, \mathrm{s} \).\[ v = 0 + 3.7 \times 20 = 74 \, \mathrm{m/s} \]

Step 2: Determine Remaining Vertical Distance

We need to calculate how far the explorer has fallen after 20 seconds.Use the formula for distance:\[ s = ut + \frac{1}{2}gt^2 \]Where \( u = 0 \), \( g = 3.7 \, \mathrm{m/s^2} \), and \( t = 20 \, \mathrm{s} \).\[ s = 0 + \frac{1}{2} \times 3.7 \times 20^2 = 740 \, \mathrm{m} \]Since the initial height is 1200 m, the remaining height is:\[ 1200 - 740 = 460 \, \mathrm{m} \]

Step 3: Determine Time Needed for Descent with PAPS

The explorer must travel the remaining 460 meters using PAPS to slow down to zero vertical velocity at the surface.Use the formula for time given constant acceleration:\[ 0 = 74 + a \cdot t \]\[ t = \frac{-74}{a} \]

Step 4: Calculate Necessary Acceleration

Use the kinematic equation for final velocity to solve for acceleration:\[ v^2 = u^2 + 2as \]Where \( v = 0 \), \( u = 74 \, \mathrm{m/s} \), and \( s = 460 \, \mathrm{m} \).\[ 0 = 74^2 + 2a \cdot 460 \]\[ a = \frac{-74^2}{2 \times 460} \approx -5.96 \, \mathrm{m/s^2} \]

Step 5: Determine Horizontal Force During Descent

The horizontal velocity of the explorer remains constant since no force is acting horizontally before the PAPS engages.The horizontal force is simply the force required to stop the horizontal movement of 33 \( \mathrm{m/s} \):\[ F_{horizontal} = ma \]\[ a = \frac{-33}{t_{PAPS}} \]We will find \(t_{PAPS}\) from the vertical component first.

Step 6: Calculate PAPS Force Magnitude and Time

Given the vertical deceleration calculated previously, apply Newton's second law to find the force:\[ F_{vertical} = ma = 150 \times 5.96 = 894 \, \mathrm{N} \]Using the time formula from vertical velocity:\[ t = \frac{-74}{-5.96} \approx 12.42 \, \mathrm{s} \]

Step 7: Determine Horizontal Force How Long It Acts

From step 5, the duration to slow to zero horizontally at this constant acceleration:\[ F_{horizontal} = ma = 150 \times \frac{33}{12.42} \approx 399 \, \mathrm{N} \]

Step 8: Final Components of PAPS Force and Time Interval

The PAPS must exert a horizontal force of approximately 399 N and a vertical force of approximately 894 N. The total time of propulsion required is approximately 12.42 seconds after the initial 20 seconds of free fall.

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In kinematics, we look at parameters such as displacement, velocity, acceleration, and time. For the problem in question, kinematics helps us analyze the motion of the explorer on Mars.

Key equations used include:

• Velocity: Used to determine the change in speed over time, given by the equation: \[ v = u + gt \]
• Distance: This helps calculate how far an object moves over time, calculated with the formula: \[ s = ut + \frac{1}{2}gt^2 \]
• Acceleration: This is the rate of change of velocity and is crucial for understanding how quickly an explorer speeds up or slows down.

Kinematics allows for the prediction and calculation of the explorer's velocity and position at different points of her descent, which is crucial for assessing how to safely slow down before reaching the Martian surface.

Newton's laws of motion

Understanding Newton's laws is critical for solving problems involving forces and motion, such as the descent of the explorer. Here’s a brief overview:

• First Law: An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. This explains why the explorer maintains a constant horizontal velocity until a force is applied.
• Second Law: Force equals mass times acceleration ( \[ F = ma \]). This pivotal formula helps us determine the force needed by the PAPS to decelerate the explorer as she approaches the surface of Mars.
• Third Law: For every action, there is an equal and opposite reaction. This concept can be observed in the way the PAPS propulsion exerts force against the explorer's movement.

Understanding these laws enables precise calculations and adjustments to ensure safe landings and operations, vital to any Martian exploration mission.

Projectile motion

Projectile motion involves the paths of objects under the influence of gravity, like the explorer stepping off a vehicle. This motion can be broken down into horizontal and vertical components, both governed by the laws of physics.

• Horizontal Motion: The explorer's horizontal speed remains constant (33 m/s) until PAPS kicks in due to a lack of horizontal forces acting initially.
• Vertical Motion: The vertical motion is accelerated by Mars' gravity, calculated from initial rest through kinematic equations. The PAPS must later counteract this to bring her to a stop.

By analyzing both components, we see that the overall path is a curve downward, dictated solely by the initial horizontal velocity and Mars' gravity. With external forces like PAPS, controlling this trajectory becomes critical for safe landings.

Mars gravity

Gravity on Mars is weaker than that on Earth, at approximately 3.7 m/s². This has significant implications for movement and force calculations.

• Reduced Gravity Effect: Lower gravitational force means objects fall slower than they would on Earth. This affects how the explorer's descent is calculated.
• Impact on Motion: The vertical and horizontal motions under Mars' gravity necessitate adjustments in force and timing. For example, reducing the descent speed requires less force than it would on Earth.
• Force Calculations: Understanding Mars' gravity is essential when determining the exact force needed to minimize velocity before landing.

Mars' unique gravitational pull affects nearly all calculations for missions, influencing everything from trajectory planning to the design of safety systems like the PAPS. Recognizing this is key to adapting Earth-based knowledge for successful Mars exploration.