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Work, Energy and Power - Collisions in One and Two Dimensions

Grade 11ICSEPhysics

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

πŸ”‘Concepts

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A collision is an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time. The law of conservation of momentum applies if the net external force F⃗ext=0\vec{F}_{ext} = 0.

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Elastic Collision: A collision in which both total momentum and total kinetic energy (K.E.K.E.) are conserved. For such collisions, the coefficient of restitution e=1e = 1.

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Inelastic Collision: A collision in which total momentum is conserved but kinetic energy is not. Some K.E.K.E. is converted into heat, sound, or internal potential energy. For these, 0≀e<10 \leq e < 1.

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Perfectly Inelastic Collision: The colliding bodies stick together after impact and move with a common velocity. Here, e=0e = 0, and the loss in K.E.K.E. is maximum.

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Coefficient of Restitution (ee): It is defined as the ratio of the relative velocity of separation to the relative velocity of approach along the line of impact: e=v2βˆ’v1u1βˆ’u2e = \frac{v_2 - v_1}{u_1 - u_2}.

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One-Dimensional Collision: The motion of the bodies stays along a single straight line before and after the impact.

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Two-Dimensional (Oblique) Collision: The colliding bodies do not move along the initial line of motion after impact. Momentum is conserved independently along both the xx and yy axes (Pix=PfxP_{ix} = P_{fx} and Piy=PfyP_{iy} = P_{fy}).

πŸ“Formulae

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

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

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

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

Ξ”K=12m1m2m1+m2(u1βˆ’u2)2(1βˆ’e2)\Delta K = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (u_1 - u_2)^2 (1 - e^2)

m1u1=m1v1cos⁑θ1+m2v2cos⁑θ2 (x-axis conservation)m_1 u_1 = m_1 v_1 \cos \theta_1 + m_2 v_2 \cos \theta_2 \text{ (x-axis conservation)}

0=m1v1sin⁑θ1βˆ’m2v2sin⁑θ2Β (y-axisΒ conservation)0 = m_1 v_1 \sin \theta_1 - m_2 v_2 \sin \theta_2 \text{ (y-axis conservation)}

πŸ’‘Examples

Problem 1:

A mass of 2Β kg2 \text{ kg} moving at 10Β m/s10 \text{ m/s} collides with a stationary mass of 3Β kg3 \text{ kg}. If the collision is perfectly inelastic, find the final common velocity and the loss in kinetic energy.

Solution:

  1. Conservation of momentum: m1u1+m2u2=(m1+m2)Vm_1 u_1 + m_2 u_2 = (m_1 + m_2)V (2)(10)+(3)(0)=(2+3)V⇒20=5V⇒V=4 m/s(2)(10) + (3)(0) = (2 + 3)V \Rightarrow 20 = 5V \Rightarrow V = 4 \text{ m/s}.
  2. Initial K.E.=12m1u12=12(2)(10)2=100Β JK.E. = \frac{1}{2} m_1 u_1^2 = \frac{1}{2}(2)(10)^2 = 100 \text{ J}.
  3. Final K.E.=12(m1+m2)V2=12(5)(4)2=40Β JK.E. = \frac{1}{2} (m_1 + m_2) V^2 = \frac{1}{2}(5)(4)^2 = 40 \text{ J}.
  4. Loss in K.E.=100βˆ’40=60Β JK.E. = 100 - 40 = 60 \text{ J}.

Explanation:

In a perfectly inelastic collision, the objects stick together. Momentum is conserved, but kinetic energy is lost to deformation and heat.

Problem 2:

In an elastic one-dimensional collision, a body of mass mm moving with velocity uu hits another identical body at rest. What are their velocities after collision?

Solution:

Given m1=m2=mm_1 = m_2 = m, u1=uu_1 = u, and u2=0u_2 = 0. Using v1=(m1βˆ’m2m1+m2)u1+(2m2m1+m2)u2v_1 = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_1 + \left( \frac{2 m_2}{m_1 + m_2} \right) u_2: v1=(mβˆ’m2m)u+0=0v_1 = \left( \frac{m - m}{2m} \right) u + 0 = 0. Using v2=(2m1m1+m2)u1+(m2βˆ’m1m1+m2)u2v_2 = \left( \frac{2 m_1}{m_1 + m_2} \right) u_1 + \left( \frac{m_2 - m_1}{m_1 + m_2} \right) u_2: v2=(2m2m)u+0=uv_2 = \left( \frac{2m}{2m} \right) u + 0 = u.

Explanation:

When two bodies of equal mass collide elastically in one dimension, they exchange their velocities. The first body comes to rest, and the second body moves with the initial velocity of the first.

Collisions in One and Two Dimensions Revision - Class 11 Physics ICSE