Animated Solution for Physics - Laws of Motion: A force $\mathbf{F}=(40\hat{i}+10\hat{j})$ N acts on a body of mass $5\text{ kg}$. If the body starts from rest its position vector $\mathbf{r}$ at time $t=10\text{ s}$ will be

Imagine a block of mass $5\text{ kg}$ resting peacefully at the origin of our coordinate system. It's just sitting there, completely undisturbed, which means its initial velocity $\mathbf{u}$ is zero.

Suddenly, a force kicks in! This isn't just a simple push in one direction; it's a 2D force vector given by $\mathbf{F}=40\hat{i}+10\hat{j}\text{ N}$. This means the force is pulling the block simultaneously along the x-axis with $40\text{ N}$ and along the y-axis with $10\text{ N}$.

According to Newton's second law of motion, a net force applied to a mass will produce an acceleration. The relationship is beautifully simple: $\mathbf{F}=m\mathbf{a}$.

To find out how our block accelerates, we just need to divide the force vector by the mass.

Notice how the acceleration vector points in the exact same direction as the force vector. The block is now speeding up, gaining $8\text{ m/s}$ of velocity in the x-direction and $2\text{ m/s}$ in the y-direction every single second!

Now, we need to find where this block ends up after exactly $10\text{ seconds}$. Since the force is constant, the acceleration is also constant. This allows us to use our trusty second equation of motion, but in its full vector glory:

Because the block started from rest, the initial velocity $\mathbf{u}$ is zero. This makes the entire $\mathbf{u}t$ term vanish completely, simplifying our lives immensely! We are left with:

Let's plug in our known values. The time $t$ is $10\text{ s}$, and we just found our acceleration vector $\mathbf{a}$.

Squaring the $10\text{ seconds}$ gives us $100$. Half of $100$ is $50$. Now, we just scale our acceleration vector by this factor of $50$:

And there we have it! Because the block started from the origin, its displacement vector $\mathbf{s}$ is exactly the same as its final position vector $\mathbf{r}$. The block has travelled along a straight line to its new position at $(400,100)$.

Always remember, if the initial velocity was non-zero and pointing in a different direction, the path wouldn't be a straight line anymore; it would be a parabola! The initial conditions dictate the shape of the trajectory.