Elastic collisions in two dimensions questions

Questions to attempt

6 questions on Elastic collisions in two dimensions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FM1-5.1 · Oblique impact of smooth elastic spheres and a smooth sphere with a fixed surface. Loss of kinetic energy due to impact.

   Q1

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   1.

   Two smooth spheres of equal radius and mass $2\,\text{kg}$ collide. At impact, $\mathbf i$ is directed along their line of centres. Their velocities are $(6\mathbf i+2\mathbf j)\,\text{m s}^{-1}$ and $(\mathbf i-3\mathbf j)\,\text{m s}^{-1}$, and $e=1/2$. Find both velocities after impact and the impulse on the first sphere.

   (5)

   (Total for Question 1 is 5 marks)

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2. FM1-5.1 · Oblique impact of smooth elastic spheres and a smooth sphere with a fixed surface. Loss of kinetic energy due to impact.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

   The unit vector $\mathbf i$ points normally towards a fixed smooth wall and $\mathbf j$ is parallel to it. A sphere arrives with velocity $(3\mathbf i-4\mathbf j)\,\text{m s}^{-1}$ and has coefficient of restitution $1/2$ with the wall. Find its velocity after impact.

   (2)

   (Total for Question 2 is 2 marks)

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3. FM1-5.1 · Oblique impact of smooth elastic spheres and a smooth sphere with a fixed surface. Loss of kinetic energy due to impact.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   Two smooth spheres have equal radius and each has mass $2\,\text{kg}$. At collision their line of centres is parallel to $\mathbf i$, and their velocities are $(5\mathbf i+4\mathbf j)\,\text{m s}^{-1}$ and $(-\mathbf i+2\mathbf j)\,\text{m s}^{-1}$. Given $e=1/3$, determine both velocities after impact and the loss of kinetic energy.

   (6)

   (Total for Question 3 is 6 marks)

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4. FM1-5.2 · Successive oblique impacts of a sphere with smooth plane surfaces.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   A sphere moves inside a right-angled corner with velocity $(8\mathbf i+6\mathbf j)\,\text{m s}^{-1}$. It strikes first the wall normal to $\mathbf i$ and then the wall normal to $\mathbf j$. Its coefficient of restitution with each wall is $1/2$. Find its velocity after the second impact and the fraction of its initial kinetic energy that remains.

   (5)

   (Total for Question 4 is 5 marks)

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5. FM1-5.2 · Successive oblique impacts of a sphere with smooth plane surfaces.

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   A sphere approaches a vertical smooth wall with velocity $(6\mathbf i+8\mathbf j)\,\text{m s}^{-1}$, where $\mathbf i$ is normal to the wall. The coefficient of restitution is $1/2$. Find its velocity and speed immediately after impact.

   (3)

   (Total for Question 5 is 3 marks)

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6. FM1-5.2 · Successive oblique impacts of a sphere with smooth plane surfaces.

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   A sphere travels inside a right-angled corner with velocity $(10\mathbf i+6\mathbf j)\,\text{m s}^{-1}$. It hits the wall normal to $\mathbf i$ with coefficient of restitution $e$, then the wall normal to $\mathbf j$ with coefficient $1/2$. After both impacts its direction is $30^\circ$ below the negative $\mathbf i$ direction. Find $e$ and the fraction of its initial kinetic energy lost.

   (6)

   (Total for Question 6 is 6 marks)

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FM1-5 Elastic collisions in two dimensions revision notes