Impact

💥 Impact

Understanding Short-Duration Collisions

Impact is a short-duration interaction between two bodies during which large contact forces act over a very small time interval.

For impact problems, the objective is generally to determine the post-impact velocities from the known pre-impact velocities, masses, geometry of contact, and coefficient of restitution.

For oblique impact, the velocity of each body is resolved into components along two mutually perpendicular directions:

Normal direction, n — along the common normal at the point of contact.
Tangential direction, t — perpendicular to the normal direction.

For smooth bodies, the impulsive contact force acts only along the normal direction. Consequently, in the absence of other impulsive forces, the tangential components of velocity remain unchanged, while the normal components are determined using appropriate linear impulse–momentum and the coefficient of restitution.

The animations in this section use representative impact problems to demonstrate how the geometry of contact determines the appropriate coordinate system and how the velocity components change during impact.

🧮 Governing Principles

Normal and Tangential Components:

For an oblique impact, the velocity of each body is resolved as

u = uₙ n + uₜ t

before impact and

v = vₙ n + vₜ t

after impact.

The normal direction is defined by the line of impact, while the tangential direction is perpendicular to it.

Conservation of Momentum Along the Line of Impact:

For two bodies A and B,

mAuAₙ + mBuBₙ = mAvAₙ + mBvBₙ

when the contact impulse is the only significant impulse acting during the impact.

Thus, the total linear momentum in the normal direction is conserved when the contact impulse is the only significant external impulse acting on the system.

For constrained motion, external impulsive reactions may be present. In such cases, linear momentum should be conserved only in a direction perpendicular to the direction of the external impulse.

Coefficient of Restitution:

The coefficient of restitution relates the relative separation velocity to the relative approach velocity along the line of impact:

e = (vBₙ − vAₙ)/(uAₙ − uBₙ)

or equivalently,

vBₙ − vAₙ = e(uAₙ − uBₙ)

where

e = 1 corresponds to a perfectly elastic impact,
e = 0 corresponds to a perfectly plastic impact,
0 < e < 1 represents a partially elastic impact.

Tangential Components:

For smooth contacting bodies, the contact impulse has no tangential component. If no other tangential impulsive force or constraint reaction acts during impact, then

vAₜ = uAₜ

vBₜ = uBₜ

The tangential components therefore remain unchanged during impact.

For an unconstrained two-body impact with negligible external impulse, these relations provide the two independent equations required to determine the unknown normal velocity components after impact. When constraints or external impulses are present, the corresponding impulse–momentum and kinematic relations must be included.

🔍 Impact Analysis Workflow

A typical oblique-impact problem can be analysed systematically:

Identify the point of contact and establish the line of impact.
Define the normal (n) and tangential (t) directions.
Resolve the initial velocities into their n–t components.
Apply the appropriate impulse–momentum relation in the normal direction, accounting for any external impulsive forces or constraint reactions.
Apply the coefficient of restitution relation along n.
Use the tangential velocity relation when the contact is smooth and no tangential impulse acts.
Apply any kinematic constraints on the bodies' motion.
Transform the resulting velocities back to the given original coordinate system.
Check the resulting directions and magnitudes of the post-impact velocities.

The animations make these steps visible by showing the instantaneous velocity vectors, the n–t coordinate system, the trajectories before and after impact, and the corresponding velocity components.

🎥 Problems Explored

1. 💥 Oblique Impact of Two Spheres

Problem: Sphere A, of mass 10 kg, collides with sphere B, of mass 2 kg, as shown. The coefficient of restitution is e = 0.5. The initial velocities and geometry of contact are specified. Determine the x- and y-components of the velocity of each sphere immediately after impact.

Concept: Application of the normal–tangential formulation of oblique impact using linear impulse–momentum and the coefficient of restitution.

🔍 Watch For:

How the line of impact is identified from the geometry of the contacting spheres.
How the n–t coordinate system is established at the instant of impact.
How the initial velocities uA and uB are resolved into normal and tangential components.
Why the tangential components remain unchanged for smooth contact.
How conservation of momentum is applied specifically along the normal direction.
How the coefficient of restitution determines the relative separation velocity.
How the resulting normal and tangential components are recombined to obtain vA and vB in the x–y coordinate system.
Observe the velocity-component inset, which continuously displays the current x- and y-components of the velocities before and after impact.
Observe how the velocity directions change abruptly at impact, while the paths remain straight both before and after the collision.

The animation also displays the angles made by the velocity vectors with the x-axis, together with the n–t directions at the instant of impact.

Source: Meriam & Kraige, Engineering Mechanics: Dynamics, 7th edition, Problem 3/262.

2. 💥 Constrained Oblique Impact of a Sphere with a Movable Carriage

Problem: A sphere A, of mass 2 kg, strikes the smooth inclined surface of a carriage B, of mass 10 kg. The coefficient of restitution is e = 0.6. The carriage is free to move horizontally on a smooth surface, while the sphere approaches the inclined face horizontally with a specified velocity. Determine the post-impact velocity of the sphere and the velocity of the carriage.

Concept: Application of the normal–tangential formulation of impact to a constrained system, using linear impulse–momentum, the coefficient of restitution, and the kinematic constraint imposed on the carriage.

🔍 Watch For:

How the inclined contact surface determines the normal (n) and tangential (t) directions at the instant of impact.
How the sphere's initial velocity is resolved into its normal and tangential components.
Why the tangential component of the sphere's velocity remains unchanged for smooth contact.
How the carriage is constrained to move only in the horizontal direction.
Why the ground reaction on the carriage introduces an external impulsive force during impact.
Why linear momentum of the complete sphere–carriage system is therefore conserved only in the x-direction, and not in the y-direction.
How the contact force on the sphere and the equal-and-opposite contact force on the carriage act along the common normal.
How the ground reaction acts vertically upward on the carriage.
How the impulse–momentum equation in the x-direction is combined with the coefficient of restitution relation along the normal direction to determine the unknown post-impact velocities.
How the resulting sphere velocity is deflected from its initial horizontal direction after impact.
How the carriage begins moving horizontally immediately after the impact.
Observe the velocity-component inset, which displays the x- and y-components of the sphere and carriage velocities before and after impact.
Observe the spring compression inset, which tracks the subsequent compression of the spring after the carriage begins moving.
Observe how the sphere follows a straight trajectory before impact and a different straight trajectory after impact.

During the impact frame, the animation explicitly shows the n–t coordinate system, the three impulsive forces acting on the system, and the statement that linear momentum of the system is conserved in the x-direction only. This provides a visual distinction between an unconstrained two-body impact and an impact involving an external constraint reaction.

After impact, the carriage moves horizontally and compresses the spring. The subsequent motion illustrates how the velocity acquired during the impact becomes the initial condition for the spring–mass motion of the carriage.

Source: Meriam, Kraige & Bolton, Engineering Mechanics: Dynamics, 8th edition, Problem 3/267.

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