Question

A particle moves so that its position as a function of time in MKS unit is \(\vec{r}=\hat{i}+4 t^{2} \hat{j}+t \hat{k}\). The trajectory of the particle is - (1) parabola (2) ellipse (3) circle (4) None of these

Short answer

The trajectory of the particle is a parabola.

Step-by-step solution

Analyze the Position Vector

The position vector \(\vec{r}\) is given by \(\vec{r} = \hat{i} + 4t^2 \hat{j} + t \hat{k}\). This means the x-component is constant at 1, the y-component varies as \(4t^2\), and the z-component varies linearly with \(t\).

Identify the Motion in the xy-plane

In the xy-plane, the x-component is constant (x = 1), and y varies as \(4t^2\). This corresponds to the parabolic equation \(y = 4t^2\).

Identify the Motion in the xz-plane

In the xz-plane, the x-component is constant (x = 1), and z varies linearly with \(t\). Therefore, \(z = t\).

Conclusion about the Trajectory

Combining the analyses, the particle follows a parabolic path in the xy-plane. Therefore, the trajectory of the particle is a parabola.

Key concepts

Position Vector

To understand the movement of a particle, it's crucial to examine its position vector. The position vector \(\backslash{}vec{r} = \backslash{}hat{i} + 4t^2 \backslash{}hat{j} + t \backslash{}hat{k}\) represents the location of the particle in a 3D space at any given time. Here, components \(\backslash{}hat{i}, \backslash{}hat{j},\) and \(\backslash{}hat{k}\) denote the x, y, and z directions, respectively.
In this function:

• The x-component is 1 (constant)
• The y-component changes with time squared \((4t^2)\)
• The z-component changes linearly with time \(t\)

This information helps us visualize the particle's motion through space as time progresses.

Parabolic Trajectory

A parabolic trajectory is a common path in physics, usually resulting from uniform acceleration in one direction and constant motion in another.
Looking at the xy-plane:

• The x-component remains constant at x = 1
• Y-component varies as \(4t^2\)

This gives us the equation for a parabola, y = 4t^2, indicating that in the xy-plane the particle traces a parabolic path.
Parabolic motion is often seen in projectile motion where the object accelerates due to gravity (in the y-direction) while maintaining constant horizontal speed.

Motion in Planes

Motion in planes can be analyzed by observing the movement in two-dimensional spaces formed by pairs of axes.

In the xy-plane:

X remains constant at x = 1

Y varies as y = 4t^2

In the xz-plane:

• X remains constant at x = 1
• Z varies linearly as z = t

Combining these observations helps us understand the overall motion of the particle.
Here, the particle’s trajectory in 3D space is seen as a combination of parabolic motion in the xy-plane and linear motion in the xz-plane, reinforcing the conclusion of a parabolic trajectory.