At t=10 s, a particle is moving from left to right with a...

At $t=10 \mathrm{s},$ a particle is moving from left to right with a speed of $5.0 \mathrm{m} / \mathrm{s}$. At $t=20 \mathrm{s}$, the particle is moving right to left with a speed of $8.0 \mathrm{m} / \mathrm{s}$. Assuming the particle's acceleration is constant, determine (a) its acceleration, (b) its initial velocity, and (c) the instant when its velocity is zero.

Key Concepts

Constant Acceleration

This concept assumes that the acceleration acting on an object remains unchanged over time. Under constant acceleration, the motion equations apply uniformly, allowing for the use of linear kinematic relationships. This simplification is essential in problems where the acceleration is constant, as it enables prediction of other kinematic quantities like displacement, velocity, and time.

Kinematic Equations

Kinematic equations describe the relationship between displacement, velocity, acceleration, and time under constant acceleration. These equations, such as v = v? + at and x = x? + v?t + 0.5at², are fundamental tools used to analyze the motion of objects. They are crucial in solving problems where one or more of these quantities are unknown, by linking them through algebraic relationships.

Sign Convention and Direction

When dealing with one-dimensional motion, it is important to assign a consistent sign convention to indicate direction. This means choosing a positive direction (e.g., rightward or upward) and a negative direction (e.g., leftward or downward). Correctly applying the sign convention is crucial for accurately determining quantities like velocity and acceleration, especially when changes in the direction of motion are involved.

Zero Velocity Condition

The condition where velocity equals zero is significant in motion analysis because it often represents a turning point in the object's path, such as the highest or lowest point in projectile motion or a transition between directions. Identifying the time when velocity is zero helps determine the overall behavior of the motion under constant acceleration and is a key step in solving problems that involve changes in the direction of motion.

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