91Ó°ÊÓ

Velocity of a particle of mass \(2 \mathrm{~kg}\) changes from \(\vec{v}_{1}=(-2 \hat{i}-2 \hat{j}) \mathrm{m} / \mathrm{s}\) to \(\vec{v}_{2}=(\hat{i}-\hat{j}) \mathrm{m} / \mathrm{s}\) after colliding with a plane surface. (1) The angle made by the plane surface with the positive \(x\)-axis is \(90^{\circ}+\tan ^{-1}\left(\frac{1}{3}\right)\) (2) The angle made by the plane surface with the positive \(x\)-axis is \(\tan ^{-1}\left(\frac{1}{3}\right)\) (3) The direction of change in momentum makes an angle \(\tan ^{-1}\left(\frac{1}{3}\right)\) with the positive \(x\)-axis. (4) The direction of change in momentum makes an angle \(90^{\circ}+\tan ^{-1}\left(\frac{1}{3}\right)\) with the plane surface.

Step by step solution

01

Calculate Change in Velocity

The initial velocity of the particle is \( \vec{v}_1 = (-2 \hat{i} - 2 \hat{j}) \, \text{m/s} \) and the final velocity is \( \vec{v}_2 = (\hat{i} - \hat{j}) \, \text{m/s} \). The change in velocity \( \Delta \vec{v} \) is given by \( \vec{v}_2 - \vec{v}_1 = (\hat{i} - \hat{j}) - (-2 \hat{i} - 2 \hat{j}) = (1 + 2) \hat{i} + (-1 + 2) \hat{j} = 3 \hat{i} + \hat{j} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity

Velocity describes the speed and direction of a moving object. In physics, it's represented as a vector, which means it has both magnitude (how fast an object is moving) and direction (the path the object follows). In our exercise, we observe the particle's velocity changing through vectors: initially from \((-2 \hat{i} - 2 \hat{j}) \text{ m/s}\) to \((\hat{i} - \hat{j}) \text{ m/s}\). The change in velocity vector helps us understand both the difference in speed and the shift in direction following a collision.

• The components \(\hat{i}\) and \(\hat{j}\) represent the particle's velocity in the horizontal and vertical directions, respectively.
• Calculation shows a velocity change of \(3 \hat{i} + \hat{j}\), indicating a jump of 3 units along the x-axis and 1 unit along the y-axis.

The change in velocity is crucial as it determines how the momentum will change, which is intrinsic to the collision analysis.

Angle with surface

The angle a surface makes with a reference line, like the positive x-axis, can greatly affect the trajectory of an object after a collision. This angle defines how the object will bounce off and in which direction its velocity components alter post-impact. The exercise presents two possibilities for angles related to the plane surface:

• First, \(90^\circ + \tan^{-1}\left(\frac{1}{3}\right)\), meaning the surface is slightly above perpendicular relative to the x-axis.
• The second option \(\tan^{-1}\left(\frac{1}{3}\right)\) indicates the surface tilts at an arced line formed by the ratio \(\frac{1}{3}\), which slightly leans below horizontal.

These angles are pivotal for understanding the redirection force exerted by the surface during a collision, causing further analysis on the resulting directional change of velocity and momentum of the particle.

Momentum direction

Momentum is a vector quantity that results from a product of an object's mass and velocity, making it dependent on both speed and direction. When momentum changes, it not only reflects in the magnitude but more importantly, in the direction post-collision. In our exercise, we explore the directionality of the momentum change:

• It's calculated as the mass velocity change resulting in the same direction as the velocity change of \(3 \hat{i} + \hat{j}\).
• The angle of momentum change can either align diagonally along \(\tan^{-1}\left(\frac{1}{3}\right)\) relative to the x-axis, aligning with the particle's new trajectory.
• Alternatively, it may reflect off the plane surface, forming an acute \(90^\circ + \tan^{-1}\left(\frac{1}{3}\right)\) angle depending on the collision specifics.

These choices capture the notion that even small variations in the angle can drastically influence the resultant path of an object after rebounding against a surface.