A student throws a set of keys vertically upward to her...

A student throws a set of keys vertically upward to her sorority sister, who is in a window $4.00 \mathrm{m}$ above. The keys are caught 1.50 s later by the sister's outstretched hand. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?

A student throws a set of keys vertically upward to her sorority sister, who is in a window $4.00 \mathrm{m}$ above. The keys are caught 1.50 s later by the sister's outstretched hand.
(a) With what initial velocity were the keys thrown?
(b) What was the velocity of the keys just before they were caught?

Key Concepts

Kinematic Equations

Kinematic equations are a set of formulas that relate displacement, initial velocity, final velocity, acceleration, and time for objects moving with constant acceleration. They provide a systematic way to solve problems involving projectile motion by allowing one to calculate any of the unknown variables when the others are known. These equations form the backbone of mechanics in analyzing both the ascent and descent phases in vertical motion.

Calculation of Initial and Final Velocities

Determining the initial velocity involves using known values such as displacement, time, and gravitational acceleration to solve the kinematic equations, setting up a relation that reflects the conditions at the start of the motion. Similarly, calculating the final velocity involves understanding how an object’s velocity changes over time under constant acceleration. Both calculations are essential in solving problems that require assessing an object’s motion at different stages of its trajectory.

Vertical Projectile Motion

This concept deals with the movement of objects that are projected straight up or down under the influence of gravity. In vertical motion, the only acceleration acting on the object is the constant gravitational acceleration, which simplifies the analysis to one dimension. Understanding this concept is crucial to analyze how displacement, velocity, and time interact when an object is thrown upward or dropped.

Gravitational Acceleration

Gravitational acceleration is a constant that represents the rate at which objects speed up or slow down due to the Earth's gravity. In most problems, it is approximated as 9.8 m/s² (or sometimes 10 m/s² for simplicity), and it acts downward regardless of the direction of the projectile’s motion. This concept is fundamental in determining the change in velocity and displacement over time in vertical motion scenarios.

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