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In physics, projectile motion refers to the motion of an object that is launched into the air and is subject only to the acceleration of gravity after it is launched. In this scenario, the explorers used a spring gun to launch a ball bearing vertically upward. This means that the projectile only moves in the vertical direction. The key factors influencing projectile motion are the initial velocity and the gravitational acceleration, which in this context is unknown and needs to be found.

• Initial velocity: The speed at which the object is launched. Here it is 15 m/sec.
• Gravity: The only force acting on the projectile after launch, pulling it back toward the planet’s surface.

For an upward launch, the object will rise, slow down due to gravity, and eventually stop at its maximum height. Here, the problem states that this maximum height is reached 20 seconds after launch. Understanding the motion allows us to calculate unknown factors, like the gravitational force in this case.

Differentiation is a method used in calculus to find the rate at which a quantity changes. In studying motion, we use differentiation to find velocity and acceleration. When given the height equation of the ball as a function of time, we differentiate it to determine the velocity function. The derivative of a function gives us the rate at which that function is changing at any given point. In our exercise, we had the height function:\[s = 15t - \frac{1}{2}g_s t^2\]Differentiating with respect to time \(t\) gives us the velocity:\[v(t) = \frac{ds}{dt} = 15 - g_s t\]This expression indicates how the velocity changes over time, which is crucial for solving problems related to motion.

Gravity is a force that attracts two bodies toward each other. On a planet, it causes objects to fall back after being projected upwards. The exercise involved an unknown gravity surface, \(g_s\), which we needed to determine. Using the fact that gravitational force impacts motion, it slows the projectile until it stops rising (maximum height).

• By setting the differentiated velocity equation to zero, we find the key time at which upward motion ceases (when the velocity equals zero).
• This point is essential because the time when velocity becomes zero (20 seconds) helps us solve for \(g_s\) using the equation derived from differentiation.

From the solved equation \(g_s t = 15\) and with \(t = 20\) \, we re-arrange and find the gravitational influence on this small planet: \(g_s = 0.75\, m/s^2\). Understanding gravity in this context illustrates how it is not a constant and can vary depending on location, particularly on celestial bodies other than Earth.