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  • Revision:Newtons laws momentum collisions

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Newton's Laws, Momentum, Collisions


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Newton's Laws

Newton's First Law: A body will continue to remain in its state of being - at constant velocity in a straight line or at rest - unless acted upon by an external resultant force.

Newton's Second Law: The resultant force a body experiences is directly proportional to the rate of change of momentum of the body, and acts in the direction of the momentum change. This law is commonly expressed in mathematics as  \small F = ma, where \small F is the resultant force acting on the body in the direction of the acceleration, \small m is the mass of the body and \small a is the acceleration of the body.

Newton's Third Law: If body A exerts a force on body B, body B exerts a force of equal magnitude and opposite direction on body A.

Linear Momentum

Momentum is a vector quantity given by the product of its mass and its velocity.

\displaystyle \mathbf{p}= m \mathbf{v}
p is the momentum in kg ms-1 or Ns
m is the mass in kg
v the velocity in ms-1


Deriving F=ma

By Newton's Second Law Force is equal to the rate of change of momentum.

\mathbf{F} = \frac{d\mathbf{p}}{dt}

Momentum is the product of mass and velocity.

\mathbf{F} = \frac{d(m\mathbf{v})}{dt}

If mass is constant this can be rewritten as:

\mathbf{F} = m\frac{d\mathbf{v}}{dt}

\frac{d\mathbf{v}}{dt} is the rate of change of velocity or acceleration. Therefore when mass is constant:

\mathbf{F} = m\mathbf{a}


Principle of Conservation of Momentum

The Principle of conservation of momentum states that when two particles collide:

total momentum before impact = total momentum after impact

Image:Collision.png

m1u1 + m2u2 = m1v1 + m2v2


Elastic collisions

In elastic collisions both kinetic energy and linear momentum are conserved. They do not exist in the real world but are idealised scenarios which physicists use to simplify the mathematical of models. For example, in kinetic molecular theory, it is assumed that the collisions between gas molecules are fully elastic. (Eliminating this assumption would make the model fiendishly difficult to manipulate.)

In an elastic collision, the final and initial velocities of the colliding particles must satisfy two conditions:

 m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2;
{\textstyle \frac{1}{2}} m_1u^2_1 + {\textstyle \frac{1}{2}} m_2u^2_2 = {\textstyle \frac{1}{2}} m_1v^2_1 + {\textstyle \frac{1}{2}} m_2v^2_2.

Solving for v_1 and v_2, we get:

 v_1 = \displaystyle \frac{u_1 (m_1 - m_2) + 2m_2u_2}{m_1+m_2};
 v_2 = \displaystyle \frac{u_2 (m_2 - m_1) + 2m_1u_1}{m_1+m_2}.

Some rather elegant results emerge from these equations. For example, if the masses of both colliding particles are equal, the particles 'exchange' velocities upon impact. Furthermore, elastic collisions have the property that

v_{1} - v_2 = u_1 - u_2.

That is to say, the relative velocity of one particle with respect to the other is reversed by the collision and the average of the momenta before and after the collision is the same for both particles. Mathematicians may view this simply as a generalization of Newton's law of restitution (in the case where e = 1.)


Inelastic collisions

In an inelastic collision only linear momentum is conserved. Kinetic energy is not conserved because as the bodies collide they suffer energy losses in the form of heat dissipation. Nevertheless total energy is always conserved. Physically speaking, inelastic collisions are the only type of collision that feature in reality.

Collisions in one dimension in which the particles coalesce (merge after collision) are inelastic. In this case, we can modify the conservation of momentum to

 m_{1} u_{1} + m_{2} u_{2} = (m_{1} + m_{2}) v_{1}.


Superelastic Collisions (Explosions)

A body of mass (m_1 + m_2) is stationary and then 'explodes' to produce two bodies, masses m_1 and m_2 moving in opposite directions with velocities v_1 and v_2. Alternatively, one is moving with velocity v_1 and the other is moving with velocity -v_2. Mathematically:

 0 = m_1 v_1 + m_2 (-v_2)
 m_2 v_2 = m_1 v_1

Kinetic Energy is clearly not conserved in this type of collision either.

Impulse

Impulse is the change in momentum of a body and is equal to the force applied to the body and the time for which it acts.

 I = mv - mu = Ft

On a force-time graph, the Impulse is the area under the graph.

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