In Fig. 9-66, particle 1 of mass m1=0.30 kg slides rightward along an x axis on a frictionless f

In Fig. 9-66, particle 1 of mass m1=0.30 kg slides rightward...

In Fig. $9-66,$ particle 1 of mass $m_{1}=0.30 \mathrm{kg}$ slides rightward along an $x$ axis on a frictionless floor with a speed of 2.0 $\mathrm{m} / \mathrm{s} .$ When it reaches $x=$ $0,$ it undergoes a one-dimensional elastic collision with stationary parti- cle 2 of mass $m_{2}=0.40 \mathrm{kg}$ . When particle 2 then reaches a wall at $x_{w}=70 \mathrm{cm},$ it bounces from the wall with no loss of speed. At what position on the $x$ axis does particle 2 then collide with particle 1 ?

In Fig. $9-66,$ particle 1 of mass $m_{1}=0.30 \mathrm{kg}$ slides rightward along
an $x$ axis on a frictionless floor with a
speed of 2.0 $\mathrm{m} / \mathrm{s} .$ When it reaches $x=$
$0,$ it undergoes a one-dimensional elastic collision with stationary parti-
cle 2 of mass $m_{2}=0.40 \mathrm{kg}$ . When particle 2 then reaches a wall at $x_{w}=70 \mathrm{cm},$ it bounces from the wall
with no loss of speed. At what position on the $x$ axis does particle 2
then collide with particle 1 ?

Key Concepts

Elastic Reflection from a Rigid Barrier

When a particle collides elastically with a rigid barrier (such as a wall), it rebounds with the same speed it had before the collision, but in the opposite direction. This concept is key in problems where subsequent collisions occur after a particle reflects off a fixed, immovable surface.

One-Dimensional Kinematics

One-dimensional kinematics deals with the analysis of motion along a straight line, focusing on parameters such as displacement, velocity, acceleration, and the time taken for specific events. This is essential for calculating the trajectories and meeting points of moving particles after collisions.

Elastic Collisions

Elastic collisions are interactions where both momentum and kinetic energy are conserved. This concept is fundamental in collision problems because it allows you to set up equations based on conservation laws to determine the final velocities of the colliding objects.

Conservation of Momentum

The conservation of momentum principle asserts that the total momentum of a system remains constant if no external forces act on it. This is crucial in collision scenarios, as it provides one of the main tools to relate the speeds of particles before and after their interaction.

Conservation of Kinetic Energy

In an elastic collision, kinetic energy is conserved in addition to momentum. This additional conservation law helps to uniquely determine the outcome of the collision, specifically the final velocities of the objects involved.

Transcript

00:01 For this problem on the topic of center of mass and linear momentum, we are shown in the figure a particle with a given mass sliding rightwards along the x -axis on a frictionless floor with a given speed.

00:14 It reaches the point x is equal to zero at which it undergoes a one -dimensional elastic collision with another stationary particle.

00:22 We are also given the mass of that particle.

00:25 The particle then reaches the wall at the distance xw, it then bounces from the wall with no loss of speed and strikes particle 1.

00:37 We want to know the position on the x -axis that particle 2 will then collide in particle 1.

00:43 Now we know we can find the velocities of the particles after the first collision, which is at x is equal to 0 and t is equal to 0 using the equations we know v1 f.

00:57 The final velocity of particle 1 is m1 minus m2 over m1 plus m2 times the initial velocity of particle 1 and these values are known so this is 0 .3 kg minus 0 .4 kg over 0 .4 kg over 0 .3 k plus 0 .4kg multiplied by the initial velocity of particle 1 which is 2 meters per second.

01:47 So calculating we get the final velocity of particle 1 to be minus 0 .29 meters per second.

01:58 The final velocity of particle 2 b2f is equal to 2 m1 over m1 plus m2 times the initial velocity of particle 1, v1i.

02:20 And this is 2 times 0 .3 kg over 0 .3 kg plus 0 .4kg.

02:38 And this again is multiplied by 2 meters per second.

02:43 And so calculating we get this to be 1 .7 meters per second.

02:48 So now we know the second particle's velocity after the collision as well as the first particle's velocity after the collision.

03:00 And this is after the first collision...