Lunar projectile motion A rock thrown vertically upward from...

Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of $24 \mathrm{~m} / \mathrm{s}$ (about $86 \mathrm{~km} / \mathrm{h}$ ) reaches a height of $s=24 t-0.8 t^{2} \mathrm{~m}$ in $t$ seconds. a. Find the rock's velocity and acceleration at time $t$. (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go? d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft?

Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of $24 \mathrm{~m} / \mathrm{s}$ (about $86 \mathrm{~km} / \mathrm{h}$ ) reaches a height of $s=24 t-0.8 t^{2} \mathrm{~m}$ in $t$ seconds.
a. Find the rock's velocity and acceleration at time $t$. (The acceleration in this case is the acceleration of gravity on the moon.)
b. How long does it take the rock to reach its highest point?
c. How high does the rock go?
d. How long does it take the rock to reach half its maximum height?
e. How long is the rock aloft?

Key Concepts

Quadratic Equations in Kinematic Problems

Many kinematic equations, especially those involving displacement under constant acceleration, are quadratic in nature. Solving these quadratic equations is essential for determining key time instants such as the duration of flight and the time taken to reach specific heights, including half of the maximum height.

Projectile Motion

Projectile motion refers to the movement of an object that is thrown or projected into the air, where the only force acting on it (in an ideal situation) is gravity. This analysis involves studying how the object’s position, velocity, and acceleration change over time under constant gravitational acceleration.

Kinematics and Differentiation

Kinematics involves the study of motion without regard to its causes and uses equations to describe displacement, velocity, and acceleration. In this context, differentiation is used to compute the velocity as the first derivative of the displacement function with respect to time, and the acceleration as the second derivative, revealing the constant gravitational force acting on the object.

Maximum Height Determination

The maximum height in projectile motion is reached when the vertical velocity of the object becomes zero. By setting the first derivative of the displacement function (velocity) equal to zero, one can solve for the time at which the object momentarily stops rising before starting its descent.

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