A small marble rolls horizontally with speed v0 off the top...

A small marble rolls horizontally with speed $v_{0}$ off the top of a platform 2.75 $\mathrm{m}$ tall and feels no appreciable air resistance. On the level ground, 2.00 $\mathrm{m}$ paping hole in the ground (Fig. $3.40 .$ ) For what range of marble speeds $v_{0}$ will the marble land in the hole?

Key Concepts

Time of Flight

The time of flight for an object launched horizontally is determined solely by its vertical motion. This time is found using the kinematic equations for free-fall, typically using the relation that the distance fallen is proportional to the square of the time elapsed, which when solving for time gives a square root dependency on the height and gravitational acceleration.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They are used to calculate variables such as displacement, velocity, and time. In the context of projectile motion, one set of these equations applies to vertical free-fall, and a simpler linear equation applies to horizontal displacement.

Horizontal Range and Inequalities

Once the time of flight is known, the horizontal range (distance traveled) can be calculated by multiplying the horizontal speed by the time of flight. When a target area such as a hole is prescribed, inequalities are set up to determine the range of speeds (and thus, distances) that allow the object to land within the specified bounds.

Projectile Motion

Projectile motion studies the trajectory of an object launched into the air, subject only to gravitational acceleration (when air resistance is negligible). This concept allows the analysis of the object's motion by separating it into horizontal and vertical components, each following different rules of motion.

Separation of Horizontal and Vertical Components

In projectile motion, the motion in the horizontal direction is at constant velocity because no acceleration acts in that plane, while the vertical motion is accelerated due to gravity. This separation simplifies the problem by allowing independent treatment of the two directions.

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