The velocity, in feet/second, of a rocket t sec into...

The velocity, in feet/second, of a rocket $t$ sec into vertical flight is given by
$$
v(t)=-3 t^{2}+192 t+120
$$
Find an expression $h(t)$ that gives the rocket's altitude, in feet, $t$ sec after liftoff. What is the altitude of the rocket 30 sec after liftoff?

Key Concepts

Evaluation of Functions

Once a mathematical expression for a physical quantity like altitude is derived through integration, evaluating this function at a specific point in time helps determine the state of the system at that moment. This step involves substitution and arithmetic calculation to provide a numerical answer that is relevant to the physical situation.

Indefinite Integration

Indefinite integration is the process of finding the antiderivative of a function. In the context of kinematics, it is used to determine a position function from a given velocity function by integrating the velocity with respect to time. This process introduces a constant of integration, which must be determined using initial conditions.

Initial Value Problems

Initial value problems involve solving differential equations by integrating and then applying initial conditions to determine any unknown constants. In kinematics, the initial altitude or position of an object is used to find the constant of integration after integrating the velocity function to obtain the altitude.

Kinematics of Motion

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. When given a velocity function, integrating it provides the position function, which describes how the altitude or position of the object changes over time.

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