Animated Solution for Physics - Laws of Motion: A boy pushes a box of mass $2\text{ kg}$ with a force $\mathbf{F}=(20\hat{\mathbf{i}}+10\hat{\mathbf{j}})\text{ N}$ on a frictionless surface. If the box was initially at rest, then ......... m is displacement along the X-axis after $10\text{ s}$.

Imagine you are standing in a perfectly smooth, frictionless ice rink. You see a box resting peacefully on the ice. Suddenly, a boy comes along and gives it a strong, angled push. This push isn't just straight forward; it's directed slightly upwards. Our mission? To figure out exactly how far this box travels horizontally along the ice in exactly ten seconds.

This problem is a beautiful demonstration of one of the most powerful principles in classical mechanics: the independence of perpendicular motions. Let's break it down step by step.

We are given a box with a mass $m=2\text{ kg}$. The boy applies a force represented as a vector: $\mathbf{F}=(20\hat{\mathbf{i}}+10\hat{\mathbf{j}})\text{ N}$.

What does this vector tell us? It means the force has two distinct components: 1. A horizontal component, $F_x=20\text{ N}$, pushing the box along the X-axis. 2. A vertical component, $F_y=10\text{ N}$, pulling the box slightly upwards along the Y-axis.

Because the surface is frictionless, we don't need to worry about the vertical force affecting any frictional resistance. The horizontal motion is governed entirely by the horizontal force.

To find out how far the box moves, we first need to know how fast it's speeding up. Enter Newton's Second Law of Motion:

We can rearrange this to solve for the acceleration vector, $\mathbf{a}$:

Let's substitute our known values into this equation:

Dividing each component by the mass ($2\text{ kg}$), we get:

This tells us that the box is accelerating at $10{\text{ m/s}}^2$ in the horizontal direction and $5{\text{ m/s}}^2$ in the vertical direction.

The question specifically asks for the displacement along the X-axis. This is where the magic of independent dimensions comes in. We can completely ignore the Y-component of the acceleration for this calculation.

We extract the horizontal parameters: - Initial horizontal velocity, $u_x=0\text{ m/s}$ (since the box was initially at rest). - Horizontal acceleration, $a_x=10{\text{ m/s}}^2$. - Time, $t=10\text{ s}$.

Now, we bring in the second equation of kinematics, which relates displacement, initial velocity, acceleration, and time:

Let's carefully substitute our isolated X-axis values into the kinematic equation:

The first term vanishes because the initial velocity is zero. We are left with:

And there we have it! The box travels a total distance of $500\text{ meters}$ along the X-axis in those ten seconds.

Notice how elegantly the vector math allowed us to separate the complex angled push into simple, manageable one-dimensional problems. This is the true power of physics!