Question

Figure 9-73 shows an overhead view of two particles sliding at constant velocity over a frictionless surface. The particles have the same mass and the same initial speed V = 4.00 m/s, and they collide where their paths intersect. An xaxis is arranged to bisect the angle between their incoming paths, such that$\mathit{\theta }\mathbf{=}\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{°}$. The region to the right of the collision is divided into four lettered sections by the xaxis and four numbered dashed lines. In what region or along what line do the particles travel if the collision is (a) completely inelastic, (b) elastic, and (c) inelastic? What are their final speeds if the collision is (d) completely inelastic and (e) elastic?

Short answer

1. After the inelastic collision, the particles will stick together and will movealong thex-axis.
2. After the collision, one particle will go along line 2 and the other along line 3.
3. After an inelastic collision, one particle will go through region B between lines 2 and x-axis andtheother through region C between line 3 and x-axis.
4. The final speed of the particles if the collision is completely inelastic is equal to.
5. The final speed of the particles if the collision is elastic is equal to 4.00 m/s .

Step-by-step solution

Listing the given quantities

Initial velocity is, v= 4.00 m/s .

The angle,$\theta =40°$ .

Understanding the concept of the law of conservation of momentum

Using the law of conservation of momentum and characteristics of different types of collision, we can find the region or line along which the particles move. Also, we can find the speed of the particles after different types of collision using this.

Formula:

The momentum of the system before collision = Momentum of the system after the collision

(a) the region or line along which the particle travels if the collision is completely inelastic.

Fromthegiven figure, we noticed that the momentum in the y direction would be zero.

Therefore, after the inelastic collision, the particles will stick together and will move along the x-axis.

(b) the region or line along which the particle travels, if the collision is elastic

In an elastic collision, the total momentum of the system should be conserved. Also, the x and y components of the momentum should be conserved. For this angle of scattering should be equal to the angle of approach.

Therefore, after the collision, one particle will go along line 2 and the other along line 3.

(c) The region or line along which the particle travels if the collision is inelastic

In inelasticcollisions, energy is not conserved.So the speedof the particles will be reduced after the collision. The x component of the totalmomentum willnot change after the collision. So y component of the total momentum will lessen, which leads to lessened velocities of the particles after the collision. This leads to smaller angles of scattering.

Therefore, one particle will go through region B between lines 2 and x-axis and the other through region C between line 3 and x-axis.

(d) Final speed of the particles if the collision is completely inelastic.

In inelasticcollisions,the total momentum is conserved. Also x component of it is conserved, and the y component is zero, which leads to the conservation of the x component of the velocity. Hence,

$$\begin{gathered} v_{fx}=v\cos \theta \\ =4\mathrm{m}/\mathrm{s}\times \cos 40° \\ =3.06\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the final speed of the particles if the collision is completely inelastic is equal to 3.06 m/s .

(e) Final speed of the particles if the collision is elastic

For this case, in the elastic collision, thespeed afterthe collisionis the sameas before.

Therefore, the final speed of the particles if the collision is elastic is equal to 4.00 m/s .