Consider an isolated system consisting of two masses, each of which has a velocity vector in two dimensions. When these two masses collide the total momentum of the system is conserved.
(a) Show that if the total momentum of the system is conserved then the component momenta are also conserved.
(b) Given that the total momentum is conserved, can we also conclude that the total kinetic energy is conserved for this collision? Explain.
If Vector A + Vector B = Vector C
then orthogonal components of the vectors also add.
If V = A + B
then
Vx = Ax + Bx
and
Vy = Ay + By
and
Vz = Az + Bz
and as many more dimensions as our tensor occupies ")
(b) NO !!!!!
It is only conserved in an elastic collision
Energy is NOT a vector for one thing.
If two blocks of the same mass come at each other from opposite directions at the same speed and stick, the resulting kinetic energy is ZERO
(By the way the momentum was conserved because it was mv-mv = 0 to start with)
To show that if the total momentum of the system is conserved, then the component momenta are also conserved, we need to understand the concept of momentum and its components.
Momentum (p) is a vector quantity defined as the product of an object's mass (m) and its velocity (v). It is given by the formula:
p = m * v
For a two-dimensional system with masses m1 and m2 and velocities v1 and v2, the total momentum (p_total) is given by the vector sum of the individual momenta:
p_total = p1 + p2
= m1 * v1 + m2 * v2
Now, let's consider the x and y components of the momenta. The x-component of the total momentum (p_total_x) is the sum of the x-component of each individual momentum:
p_total_x = p1_x + p2_x
= m1 * v1_x + m2 * v2_x
Similarly, the y-component of the total momentum (p_total_y) is the sum of the y-component of each individual momentum:
p_total_y = p1_y + p2_y
= m1 * v1_y + m2 * v2_y
Now, if the total momentum (p_total) of the system is conserved, it means that its magnitude and direction remain constant before and after the collision. Mathematically, we can write it as:
p_total_initial = p_total_final
Now, let's examine the x-component of the total momentum:
p_total_x_initial = m1 * v1_x_initial + m2 * v2_x_initial
p_total_x_final = m1 * v1_x_final + m2 * v2_x_final
If p_total_x_initial = p_total_x_final, it implies that the x-component of the total momentum is also conserved. Similarly, by applying the same logic to the y-component, we can show that the y-component of the total momentum is conserved too. Hence, we can conclude that if the total momentum of the system is conserved, then the component momenta are also conserved.
Now, moving on to the second part of the question:
In general, the total kinetic energy of a system is NOT conserved in a collision. During a collision, energy can be transferred between objects in various forms, such as heat, sound, or deformation. This transfer of energy can cause a change in the total kinetic energy of the system.
Therefore, given that the total momentum is conserved, we cannot conclude that the total kinetic energy is conserved for this collision. In order to determine the change in kinetic energy, we need to consider the specific details of the collision, the forces involved, and any potential energy transformations.