91Ó°ÊÓ

In the context of this problem, kinematics helps us explore how the lunar lander moves as it descends towards the moon's surface. Essentially, kinematics involves the study of motion, allowing us to understand elements like displacement, velocity, and acceleration. For the lunar lander, we observe its height at any given time with the equation: \(y(t)=b-ct+dt^{2}\). This equation breaks down into:

• The initial height \(b=800 \text{ m}\)
• A constant term \(ct\) where \( c = 60.0 \text{ m/s} \), which indicates the initial downward velocity's influence over time
• An acceleration term \( dt^2 \) where \( d = 1.05 \text{ m/s}^2 \), representing the influence of an extra force or change in speed over time

Understanding these components allows us to track how the lander falls and predict its actions at different times.

To find the velocity of the lander at any given time, we need to derive the height equation \(y(t)=b-ct+dt^{2}\). In physics, the derivative signifies the rate of change. Here, we derive to find how the height changes over time—essentially calculating velocity.The derivative of a function gives us its rate of change. Using this principle:\[\frac{dy}{dt} = v(t) = -c + 2dt\] In this instance:

• The constant \(-c\) represents the initial constant speed influencing descent.
• The term \(2dt\) modifies this as time passes, reflecting acceleration's effect on increasing velocity.

Through derivation, we're equipped to find the exact speed at any specific point in time, such as initially (\(t=0\)) or just before touching the lunar surface.

Solving when the lunar lander reaches the surface requires a solution to the quadratic equation derived from \(y(t) = 0\). Rearranging gives us:\[dt^2 - ct + b = 0\]This is recognizable as a standard quadratic equation, and we can solve it using the quadratic formula:\[t = \frac{-(-c) \pm \sqrt{(-c)^2 - 4 \cdot d \cdot b}}{2 \cdot d}\]The quadratic formula is particularly useful in physics for solving motion-related problems where the position involves time squares:

• \((-c)\) contributes to the linear portion of the solution.
• \(d\) factors into the acceleration.
• \(b\) adjusts the formula for the initial conditions.

Choosing the positive root from this formula answers when the lander's height equals zero, marking lunar surface contact.

Above, we've calculated the time it takes for the lander to descend using the quadratic formula, but finding velocities is an equally crucial task. The velocity function \(v(t) = -c + 2dt\) helps us plug and chug:When \(t=0\), \[v(0) = -c + 2d(0) = -60.0 \text{ m/s}\]This represents how fast the lunar lander is moving downward initially.Simply substitute the time \(t\), found from solving the quadratic equation, back into the velocity function:\[v(t) = -c + 2dt\]This calculation determines how the speed has increased by the time just before impact. It encompasses:

• The constant velocity's effect \((-c)\).
• Acceleration over time \((2dt)\).

This process allows you to precisely measure the lander's velocity moments before touchdown. Understanding these calculations can simplify the analysis of descending objects under gravity.