A 2.00-kg box is moving to the right with speed 9.00 m/s on a horizontal, frictionless surface. At t = 0 a horizontal force is applied to the box. The force is directed to the left and has magnitude F(t) = (6.00 N/s2)t2. (a) What distance does the box move from its position at t = 0 before its speed is reduced to zero?

Verified step by step guidance

1. Identify the given values: mass of the box (m) = 2.00 kg, initial velocity (v_i) = 9.00 m/s, and the force function F(t) = (6.00 N/s^2)t^2.
2. Understand that the force is variable and depends on time. The force function F(t) = (6.00 N/s^2)t^2 implies that the force increases with the square of time.
3. Use Newton's second law, F = ma, to set up the equation of motion. Since the force is in the opposite direction of the velocity, the acceleration a(t) = -F(t)/m = -(6.00 N/s^2)t^2 / 2.00 kg.
4. Integrate the acceleration function a(t) to find the velocity function v(t). Remember to use the initial condition v(0) = 9.00 m/s to solve for the constant of integration.
5. Set v(t) = 0 and solve for t to find the time at which the velocity becomes zero. Then, integrate the velocity function v(t) to find the displacement function x(t), using the initial condition x(0) = 0. Evaluate x(t) at the time when the velocity is zero to find the distance moved before the box stops.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, the applied force affects the box's acceleration, which is crucial for determining how its velocity changes over time.

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, we need to integrate the acceleration (derived from the force) to find the velocity as a function of time, and then integrate again to find the distance traveled before the box comes to a stop.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the applied force will reduce the kinetic energy of the box until it comes to a stop. Understanding this principle allows us to relate the force applied over a distance to the change in the box's speed, facilitating the calculation of the distance traveled before it stops.