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Work, Energy, and Power - Elastic and Inelastic Collisions

Grade 11CBSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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Collision is an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time.

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In all types of collisions (elastic and inelastic), the total linear momentum is conserved: βˆ‘pβƒ—initial=βˆ‘pβƒ—final\sum \vec{p}_{initial} = \sum \vec{p}_{final}, provided no external force acts on the system.

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Elastic Collision: A collision in which there is no loss of kinetic energy. Both momentum and kinetic energy are conserved (Kinitial=KfinalK_{initial} = K_{final}).

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Inelastic Collision: A collision in which linear momentum is conserved, but kinetic energy is not conserved. Kinetic energy is transformed into other forms like heat, sound, or deformation energy (Kinitial>KfinalK_{initial} > K_{final}).

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Perfectly Inelastic Collision: A special case of inelastic collision where the two colliding bodies stick together and move with a common velocity after the collision.

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Coefficient of Restitution (ee): A measure of the degree of elasticity of a collision, defined as the ratio of the relative velocity of separation to the relative velocity of approach. e=1e = 1 for elastic, e=0e = 0 for perfectly inelastic, and 0<e<10 < e < 1 for inelastic collisions.

πŸ“Formulae

m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

\frac{1}{2} m_1 u_1^2 + rac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \text{ (For Elastic Collisions only)}

v1=(m1βˆ’m2m1+m2)u1+(2m2m1+m2)u2v_1 = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_1 + \left( \frac{2m_2}{m_1 + m_2} \right) u_2

v2=(2m1m1+m2)u1+(m2βˆ’m1m1+m2)u2v_2 = \left( \frac{2m_1}{m_1 + m_2} \right) u_1 + \left( \frac{m_2 - m_1}{m_1 + m_2} \right) u_2

e=v2βˆ’v1u1βˆ’u2e = \frac{v_2 - v_1}{u_1 - u_2}

Ξ”K=12m1m2m1+m2(u1βˆ’u2)2Β (LossΒ inΒ K.E.Β forΒ perfectlyΒ inelasticΒ collision)\Delta K = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (u_1 - u_2)^2 \text{ (Loss in K.E. for perfectly inelastic collision)}

πŸ’‘Examples

Problem 1:

A body of mass m1=5Β kgm_1 = 5 \text{ kg} moving with a velocity of 10Β m/s10 \text{ m/s} collides with another body of mass m2=10Β kgm_2 = 10 \text{ kg} at rest. If the collision is perfectly inelastic, calculate the common velocity and the loss in kinetic energy.

Solution:

  1. Using conservation of momentum: m1u1+m2u2=(m1+m2)Vm_1 u_1 + m_2 u_2 = (m_1 + m_2)V (5)(10)+(10)(0)=(5+10)Vβ€…β€ŠβŸΉβ€…β€Š50=15Vβ€…β€ŠβŸΉβ€…β€ŠV=5015=3.33Β m/s(5)(10) + (10)(0) = (5 + 10)V \implies 50 = 15V \implies V = \frac{50}{15} = 3.33 \text{ m/s}.
  2. Initial Kinetic Energy: Ki=12m1u12=12(5)(10)2=250Β JK_i = \frac{1}{2} m_1 u_1^2 = \frac{1}{2}(5)(10)^2 = 250 \text{ J}.
  3. Final Kinetic Energy: Kf=12(m1+m2)V2=12(15)(3.33)2β‰ˆ83.33Β JK_f = \frac{1}{2} (m_1 + m_2) V^2 = \frac{1}{2}(15)(3.33)^2 \approx 83.33 \text{ J}.
  4. Loss in Kinetic Energy: Ξ”K=Kiβˆ’Kf=250βˆ’83.33=166.67Β J\Delta K = K_i - K_f = 250 - 83.33 = 166.67 \text{ J}.

Explanation:

In a perfectly inelastic collision, the two bodies stick together. The total momentum before the collision equals the total momentum after. The kinetic energy is not conserved, and the 'missing' energy is converted into heat or sound.

Problem 2:

Show that in a one-dimensional elastic collision of two bodies of equal mass (m1=m2m_1 = m_2), the bodies exchange their velocities after collision.

Solution:

Given m1=m2=mm_1 = m_2 = m. From the velocity formula for v1v_1: v1=(mβˆ’mm+m)u1+(2mm+m)u2=0β‹…u1+2m2mu2=u2v_1 = \left( \frac{m - m}{m + m} \right) u_1 + \left( \frac{2m}{m + m} \right) u_2 = 0 \cdot u_1 + \frac{2m}{2m} u_2 = u_2. From the velocity formula for v2v_2: v2=(2mm+m)u1+(mβˆ’mm+m)u2=2m2mu1+0β‹…u2=u1v_2 = \left( \frac{2m}{m + m} \right) u_1 + \left( \frac{m - m}{m + m} \right) u_2 = \frac{2m}{2m} u_1 + 0 \cdot u_2 = u_1. Thus, v1=u2v_1 = u_2 and v2=u1v_2 = u_1.

Explanation:

When two identical masses collide elastically in one dimension, the first mass takes the initial velocity of the second, and the second mass takes the initial velocity of the first. This is often observed in Billiards.

Elastic and Inelastic Collisions - Revision Notes & Key Formulas | CBSE Class 11 Physics