For free fall near the surface of a planet where the...

For free fall near the surface of a planet where the acceleration due to gravity has a constant magnitude of $g$ length-units/sec', Equation (1) in Exercise 125 takes the form $$ s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0} $$ where $s$ is the body's height above the surface. The equation has a minus sign because the acceleration acts downward, in the direction of decreasing $s$. The velocity $v_{0}$ is positive if the object is rising at time $t=0$ and negative if the object is falling. Instead of using the result of Exercise $125,$ you can derive Equation (2) directly by solving an appropriate initial value problem. What initial value problem? Solve it to be sure you have the right one, explaining the solution steps as you go along.

For free fall near the surface of a planet where the acceleration due to gravity has a constant magnitude of $g$ length-units/sec', Equation (1) in Exercise 125 takes the form
$$
s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}
$$
where $s$ is the body's height above the surface. The equation has a minus sign because the acceleration acts downward, in the direction of decreasing $s$. The velocity $v_{0}$ is positive if the object is rising at time $t=0$ and negative if the object is falling.

Instead of using the result of Exercise $125,$ you can derive Equation (2) directly by solving an appropriate initial value problem. What initial value problem? Solve it to be sure you have the right one, explaining the solution steps as you go along.

Key Concepts

Differential Equations in Kinematics

In kinematics, the motion of bodies under constant acceleration is described by differential equations. The second derivative of position with respect to time gives the acceleration, and in free fall this acceleration is constant (-g). This relationship forms the basis for deriving the equations of motion through integration.

Integration of Acceleration

Integration is used to transition from acceleration to velocity and from velocity to position. Starting with the equation s''(t) = -g, integrating once yields the velocity function v(t) = -gt + v?. A further integration gives the displacement function s(t) = -½gt² + v?t + s?, which is the kinematic equation for an object in free fall.

Free Fall Motion

Free fall motion refers to the movement of an object under the influence only of gravity, where the acceleration due to gravity (g) is constant. In this context, the direction of acceleration is downward (typically taken as negative when upward is defined as positive), which is essential for setting up the equations of motion correctly.

Initial Value Problem

An initial value problem involves solving a differential equation with given initial conditions. In the case of free fall, the second order differential equation s''(t) = -g is solved subject to the initial conditions s(0) = s? and s'(0) = v?, representing the object's initial height and initial velocity respectively.

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