Animated Solution for Physics - Work, Energy, and Power: A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is

Imagine you are pushing a heavy block across a floor. The harder you push, and the farther you push it, the more energy you transfer to the block. This fundamental idea is beautifully captured by the Work-Energy Theorem, which is the star of this problem.

We are given a particle that starts from rest, meaning its initial velocity $u=0$. Consequently, its initial kinetic energy $K_i$ is exactly $0$ J. The particle is subjected to a variable force, and we are provided with a Force-Displacement ($F-x$) graph. Our goal is to find the kinetic energy of the particle after it has travelled a distance of $3$ m.

The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy:

Since $K_i=0$, the final kinetic energy $K_f$ is simply equal to the total work done $W$. But how do we find the work done from a graph? Mathematically, work is the integral of force with respect to displacement ($W=\int Fdx$). Geometrically, this integral is exactly the area under the Force-Displacement curve.

To find the total area under the curve from $x=0$ to $x=3$, we can break the complex shape into three simple, manageable geometric figures:

1. The First Rectangle ($A_1$): From $x=0$ to $x=2$, the force is constant at $2$ N. This forms a rectangle with a width of $2$ m and a height of $2$ N.

2. The Second Rectangle ($A_2$): From $x=2$ to $x=3$, we can draw a smaller rectangle at the base. Its width is $1$ m (from $2$ to $3$) and its height is $2$ N.

3. The Triangle ($A_3$): Sitting right on top of the second rectangle (from $x=2$ to $x=3$) is a triangle. Its base is $1$ m, and its height goes from $2$ N to $3$ N, which is a height of $1$ N.

Now, we simply sum up these individual areas to find the total work done:

Since the final kinetic energy equals the total work done, we have:

This problem was straightforward because the force was always positive, meaning it was constantly adding kinetic energy to the particle. However, always be cautious! If the force graph dips below the x-axis, the area becomes negative. Negative work means the force is opposing the motion, which would decrease the particle's kinetic energy. Always respect the sign conventions in graphical physics problems!