According to one model that takes into account air...

According to one model that takes into account air resistance, the acceleration $a(t)$ (in $\mathrm{m} / \mathrm{s}^{2}$ ) of a skydiver of mass $m$ in free-fall satisfies $$ a(t)=-9.8+\frac{k}{m} v(t)^{2} $$ where $v(t)$ is velocity (negative since the object is falling) and $k$ is a constant. Suppose that $m=75 \mathrm{~kg}$ and $k=0.24 \mathrm{~kg} / \mathrm{m}$. (a) What is the skydiver's velocity when $a(t)=-4.9 ?$ (b) What is the skydiver's velocity when $a(t)=0 ?$ (This velocity is the terminal velocity, the velocity attained when air resistance balances gravity and the skydiver falls at a constant speed.)

According to one model that takes into account air resistance, the acceleration $a(t)$ (in $\mathrm{m} / \mathrm{s}^{2}$ ) of a skydiver of mass $m$ in free-fall satisfies
$$
a(t)=-9.8+\frac{k}{m} v(t)^{2}
$$
where $v(t)$ is velocity (negative since the object is falling) and $k$ is a constant. Suppose that $m=75 \mathrm{~kg}$ and $k=0.24 \mathrm{~kg} / \mathrm{m}$.
(a) What is the skydiver's velocity when $a(t)=-4.9 ?$
(b) What is the skydiver's velocity when $a(t)=0 ?$ (This velocity is the terminal velocity, the velocity attained when air resistance balances gravity and the skydiver falls at a constant speed.)

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Key Concepts

Newton's Second Law

This law states that the total force acting on an object is equal to the mass of the object multiplied by its acceleration. In problems involving motion, it allows us to set up equations that relate forces (such as gravity and air resistance) to the acceleration of the object and, ultimately, to its velocity.

Air Resistance

Air resistance, or drag, is the force that opposes the motion of an object through a fluid, in this case air. This force acts in the opposite direction of the object's velocity and increases as the speed increases, affecting both the acceleration and deceleration of the object during free fall.

Quadratic Drag Force

Quadratic drag describes a type of air resistance where the drag force is proportional to the square of the object's speed. This model is often used for objects moving at high speeds or those with large surface areas relative to their mass, and it plays a crucial role in the dynamics of free-falling bodies.

Terminal Velocity

Terminal velocity is reached when the upward force due to air resistance equals the downward gravitational force, resulting in zero net acceleration. At this point, the object continues to fall at a constant speed, which is determined by the balance between gravity and the drag force.

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