### Collision of Particles in 3-D SpaceConsider a collision between two particles \( p \) and \( q \) in 3-D space \(\mathbb{R}^3\). Suppose that the mass of \( p \) is \( a > 0 \) kilograms and the mass of \( q \) is \( b > 0 \) kg. Let \( \vec{u}_i \) and \( \vec{v}_i \in \mathbb{R}^3 \) be the constant initial velocities of \( p \) and \( q \), respectively. Let \( \vec{u}_f \) and \( \vec{v}_f \in \mathbb{R}^3 \) be the constant (final) velocities of \( p \) and \( q \), respectively, after the collision; and assume \( \vec{u}_f \neq \vec{u}_i \).1. **Write down** an equation describing the physical fact that the total momentum of the particles \( p \) and \( q \) after the collision is equal to the total momentum of \( p \) and \( q \) before the collision.2. The **impulse** of \( p \) associated with the change in velocity of \( p \) is given by \[ \vec{I} := a (\vec{u}_f - \vec{u}_i). \] The analogous **impulse** of \( q \) is \[ \vec{J} := b (\vec{v}_f - \vec{v}_i). \]Using Part (1), **prove** that the vectors \( \vec{I} \) and \( \vec{J} \) are anti-parallel.3. Assume further that the collision between \( p \) and \( q \) is **elastic**. So, the total kinetic energy of the particles \( p \) and \( q \) after the collision is equal to the total kinetic energy of \( p \) and \( q \) before the collision. **Write down** an equation that precisely describes this conservation of kinetic energy.

Given that:Mass of p=aMass of q=bInitial velocity of p=ui→Initial velocity of q=vi→Final velocity of…

Consider a collision between two particles p and q in 3-D space R³. Suppose that the mass of p is a > 0 kilograms and the mass of q is b> 0 kg. Let tively. Let and 7, E R³ be the constant initial velocities of p and q, respec- , and 7, R³ be the constant (final) velocities of p and q, respectively, after the collision; and assume ₁. (1) Write down an equation describing the physical fact that the total momentum of the particles p and q after the collision is equal to the the total momentum of p and q before the collision. (2) The impulse of p associated with the change in velocity of p is given by 7:= a (₁-₁). f The analogous impulse of q is F:= b (√ƒ - Vi). f Using Part (1), prove that the vectors and I are are anti-parallel. (3) Assume further that the collision between p and q is elastic. So, the total kinetic energy of the particles p and q after the collision is equal to the the total kinetic energy of p and q before the collision. Write down an equation that precisely describes this conservation of kinetic energy

Consider a collision between two particles p and q in 3-D space R³. Suppose that the mass of p is a > 0 kilograms and the mass of q is b> 0 kg. Let tively. Let and 7, E R³ be the constant initial velocities of p and q, respec- , and 7, R³ be the constant (final) velocities of p and q, respectively, after the collision; and assume ₁. (1) Write down an equation describing the physical fact that the total momentum of the particles p and q after the collision is equal to the the total momentum of p and q before the collision. (2) The impulse of p associated with the change in velocity of p is given by 7:= a (₁-₁). f The analogous impulse of q is F:= b (√ƒ - Vi). f Using Part (1), prove that the vectors and I are are anti-parallel. (3) Assume further that the collision between p and q is elastic. So, the total kinetic energy of the particles p and q after the collision is equal to the the total kinetic energy of p and q before the collision. Write down an equation that precisely describes this conservation of kinetic energy