A 2.00 kg box is moving to the right with speed 9.00 m / s on a horizontal, friction less surf

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A 2.00 kg box is moving to the right with speed 9.00 m / s...

A $2.00 \mathrm{~kg}$ box is moving to the right with speed $9.00 \mathrm{~m} / \mathrm{s}$ on a horizontal, friction less surface. At $t=0$ a horizontal force is applied to the box. The force is directed to the left and has magnitude $F(t)=\left(6.00 \mathrm{~N} / \mathrm{s}^{2}\right) t^{2} .$ (a) What distance does the box move from its position at $t=0$ before its speed is reduced to zero? (b) If the force continues to be applied, what is the speed of the box at $t=3.00 \mathrm{~s} ?$

Key Concepts

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is fundamental in analyzing the motion of objects under applied forces, determining how a given force (whether constant or variable) affects the motion and acceleration of an object.

Variable Acceleration and Time-Dependent Forces

When the force applied to an object is not constant but varies with time, the resulting acceleration also becomes a function of time. In such cases, the standard kinematic equations for constant acceleration no longer apply directly, and the acceleration must be expressed as a time-dependent function derived from F = ma.

Integration in Kinematics

In problems involving variable acceleration, calculus becomes a critical tool. The velocity function can be obtained by integrating the acceleration function with respect to time, while the displacement (position) function is found by integrating the velocity function with respect to time. Both integrations require careful application of initial conditions to accurately capture the object's motion.

Initial Conditions in Motion Analysis

Initial conditions, such as initial velocity and initial position, play a crucial role when integrating acceleration to derive velocity and displacement. These conditions ensure that the integrated functions accurately reflect the object's motion from the start of the interaction with the applied force.

Solving for Specific Times (e.g., Stopping Time)

In dynamics problems, it is often necessary to find the specific time at which certain events occur, such as when an object comes to a stop. This involves setting the velocity function equal to zero and solving for time, which is an application of both algebra and calculus in kinematics.

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