A ballast is dropped from a stationary hot-air balloon that...

A ballast is dropped from a stationary hot-air balloon that is hovering at an altitude of $400 \mathrm{ft}$. The velocity of the ballast after $t$ sec is $(-32 t+4) \mathrm{ft} / \mathrm{sec}$. a. Find the height $h(t)$ of the ballast from the ground at time $t$ b. When will the ballast strike the ground? c. Find the velocity of the ballast when it hits the ground.

A ballast is dropped from a stationary hot-air balloon that is hovering at an altitude of $400 \mathrm{ft}$. The velocity of the ballast after $t$ sec is $(-32 t+4) \mathrm{ft} / \mathrm{sec}$.
a. Find the height $h(t)$ of the ballast from the ground at time $t$
b. When will the ballast strike the ground?
c. Find the velocity of the ballast when it hits the ground.

Key Concepts

Integration to Determine Position

When provided with a velocity function, integration is used to derive the position (height) function. This process is central in kinematics, as it converts a rate of change (velocity) into a cumulative measure (position) with respect to time.

Application of Initial Conditions

Initial conditions are essential for determining the constant of integration that arises from integrating the velocity function. They ensure that the solution for the position function accurately reflects the starting conditions of the motion.

Kinematics of Free Fall

Free fall under gravity serves as an important example in kinematics where the acceleration is constant. The equations of motion, such as those relating height, velocity, and time, are derived based on this constant acceleration, typically incorporating a negative sign due to the downward direction.

Solving Quadratic Equations in Motion Problems

When the resulting position function is set equal to a specific value, such as zero for ground impact, the equation often takes a quadratic form in time. Solving this quadratic is crucial to determine events like the time of impact.

Evaluating Instantaneous Velocity

Once the time of a particular event (such as the moment of impact) is determined from the position function, the velocity function is used to calculate the instantaneous velocity at that moment. This step connects the time-dependent motion analysis with the practical application of determining speed and direction at a given event.

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