Engineering Mechanics : Collisions & Virtual Work

• Dimensions of linear momentum is M.L.T^–1 and units kg.m.s^–1.
• The direction of the momentum vector is identical to that of the velocity vector.                  

Newton’s second law in terms of linear momentum can be expressed in this way

• The time rate of change of linear momentum of the object (system) is the resultant force on an object (system).
• If the system is an isolated one, a law regarding conservation of linear momentum can be formulated.

1. Impulse and Momentum

• Change in momentum can be simply called as an impulse. The force of interaction might vary with time in a collision between two particles.
• Suppose the force is present for a short amount of time, zero before clock time t_i and zero after clock time t_f and it has a comparatively high value at maximum.
• The time elapsed for the interaction is approximately, ∆t = t_f – t_i .

From the figure above, it can be seen that force varies over a relatively short time. The magnitude of the impulse is given by the area under the force curve.

• We must integrate, over the elapsed time for the interaction, to find the change in momentum:

• Change in momentum is known as the impulse and is depicted by the symbol vector J.

• Impulse has the same units and dimensions as momentum and is also a vector.
• The magnitude of impulse is equal to the area under the force curve between the two clocked times, that is, over the duration of the collision.
• The impulse vector's direction is the same as that of the direction of the change in momentum vector.

As per Newton's Second Law, the net force equals the acceleration times mass of the object,

Substitution of this can be done into the equation for impulse,

The difference between the velocities at the starting and ending times is the change in velocities,

                                                                   Δv= v₂-v₁

Thus, the formula for impulse becomes,

The above equation is called the impulse-momentum theorem. It can also be seen as the change in momentum of an object in a certain time interval equals the impulse of the net force that acts for the particular time interval on the object. With the use of this formula, changes in momentum can be related to the forces that are applied to cause the change. There is a change in momentum resulting from the force that is applied for a the specified time duration.

1. Conservation of Momentum
   

• During the interaction of two objects, like that in a collision, forces may be exerted on each other.
• The forces the objects exert on each other can be considered to be a part of a closed or isolated system and in that case, the forces involved are internal forces.
• If any external force affects the system, it is called external force.
• As per Newton's Third Law, if there are no external forces, the internal forces that act between two objects have opposite directions and equal magnitude. Assume the two objects as A and B, the forces they exert on each other are,

• When collision occurs, these forces act for the same amount of time. Consider the collision to begin at time t₁ and end at time t₂, which makes the time duration of the collision as Δt , and the object A experiences the impulse equal to,

• On substituting these impulse equations in place of the forces we get,

Hence the impulse momentum equation becomes equivalent to -

• In this equation, P_A1  means the momentum of object A at time t₁, P_A2 means the momentum of object A at time t₂, P_B1 means the momentum of object B at time t₁, and P_B2 means the momentum of object B at time t₂.
• The equation can be rearranged to put all of the terms for time t₁ on one side, and terms for time t₂ on the other,

• Without the presence of any external force, the total momentum of the system remains constant. This is referred to as conservation of momentum. 
• The total momentum labeled as P, for any number of objects equals

Even if the momenta of the individual objects change, the total momentum will remain the same when no external force is acting on the bodies.

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