91Ó°ÊÓ

Kinetic friction is the force that opposes the motion of an object as it slides over a surface. In this exercise, kinetic friction comes into play when the block moves on a surface with a friction coefficient given by \(\mu_k = 0.2\). This coefficient is a measure of how slippery or sticky the surface is. The force of kinetic friction \(F_f\) is calculated using the equation \(F_f = \mu_k \cdot w\), where \(w\) is the weight of the block. The weight is 10 lb, resulting in a frictional force of 2 lb. This force acts in the opposite direction to the block's movement, slowing it down.

Newton's Second Law plays a critical role in understanding how forces affect motion. Newton’s Second Law states that the acceleration \(a\) of an object is directly proportional to the net force \(F_R\) acting upon it and inversely proportional to its mass \(m\). Mathematically, it is described by the equation \(F = ma\).

In this scenario, the resultant force \(F_R\) is the difference between the applied force \(F = 8t^2\) and the frictional force \(F_f\). To find acceleration, we use the mass of the block, which is the weight divided by gravitational acceleration \(g = 32.2\, \text{ft/s}^2\). Thus, the mass \(m\) is 0.3106 lb·s²/ft. By understanding this law, we can predict how the block's velocity will change over time due to these forces.

The calculation of acceleration involves determining how rapidly a block speeds up or slows down due to the forces acting on it. We use the relation from Newton's Second Law, \(a = F_R/m\). With an applied force described by \(F = 8t^2\), and a frictional force of 2 lb, the net force \(F_R\) becomes \(8t^2 - 2\).

Substituting into the equation for acceleration, we get \(a(t) = \frac{8t^2 - 2}{0.3106}\). This function describes how the block accelerates over time based on the balance of the forces. Calculating acceleration is essential as it directly informs us of how the velocity will change as time progresses.

Velocity determination is about computing how fast the block moves at a certain point in time. The concept of velocity includes both the speed and direction of motion. The formula used here is \(v = v_0 + a \cdot t\), where \(v_0\) is the initial velocity and \(a\) is the acceleration.

The problem specifies that the block begins with a velocity of \(v_0 = 4\, \text{ft/s}\). To find the velocity at \(t = 2\, \text{s}\), we already calculated the time-dependent acceleration. Substituting this and \(t = 2s\) into the velocity formula helps determine the block's velocity. This final value provides insight into the motion of the block after all forces and the initial condition have acted on it.