The acceleration of a rocket traveling upward is given by...

The acceleration of a rocket traveling upward is given by $a=(6+0.02 s) \mathrm{m} / \mathrm{s}^{2},$ where $s$ is in meters. Determine the time needed for the rocket to reach an altitude of $s=100 \mathrm{~m}$. Initially, $v=0$ and $s=0$ when $t=0$.

Key Concepts

Application of Initial Conditions

Initial conditions, such as starting position and velocity, are essential in solving differential equations that describe motion. They allow for the determination of the constants of integration, ensuring that the resulting equations accurately reflect the physical scenario described.

Solving for Time Using the Displacement Function

After obtaining the displacement as a function of time through successive integrations, determining the time for a specific displacement involves solving the resulting equation. This often requires algebraic manipulation, and in cases where the displacement function is not linear, can lead to quadratic or higher-order equations that must be solved for time.

Variable Acceleration Kinematics

Kinematics examines how objects move, and when acceleration is not constant, it becomes necessary to express acceleration as a function of time or position. This concept is key because it requires using calculus to relate acceleration to velocity and displacement.

Integration for Velocity and Displacement

When acceleration varies, integration is the primary tool used to derive the velocity function from the acceleration function, and subsequently, the displacement function from the velocity function. This method underpins the systematic derivation of motion equations from given acceleration profiles.

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