Spheres A(mass 0.020 kg), B(mass 0.030 kg), and C (mass 0.050 kg ) are approaching the origin

Spheres A(mass 0.020 kg), B(mass 0.030 kg), and C (mass...

Spheres $A(\operatorname{mass} 0.020 \mathrm{~kg}), B(\operatorname{mass} 0.030 \mathrm{~kg}),$ and $C$ (mass $0.050 \mathrm{~kg}$ ) are approaching the origin as they slide on a frictionless air table. The initial velocities of $A$ and $B$ are given in Fig. $\mathrm{P8.69.}$. All three spheres arrive at the origin at the same time and stick logether. (a) What must the $x$ - and $y$ -components of the initial velocity of $C$ be if all three objects are to end up moving at $0.50 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction after the collision? (b) If $C$ has the velocity found in part (a), what is the change in the kinetic energy of the system of three spheres as a result of the collision?

Spheres $A(\operatorname{mass} 0.020 \mathrm{~kg}), B(\operatorname{mass} 0.030 \mathrm{~kg}),$ and $C$ (mass
$0.050 \mathrm{~kg}$ ) are approaching the origin as they slide on a frictionless air table. The initial velocities of $A$ and $B$ are given in Fig. $\mathrm{P8.69.}$. All three spheres arrive at the origin at the same time and stick logether. (a) What must the $x$ - and $y$ -components of the initial velocity of $C$ be if all three objects are to end up moving at $0.50 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction after the collision? (b) If $C$ has the velocity found in part (a), what is the change in the kinetic energy of the system of three spheres as a result of the collision?

Key Concepts

Conservation of Linear Momentum

In a collision, regardless of whether it is elastic or inelastic, the total linear momentum of the system remains constant if no external forces act. For problems involving two or three dimensions, momentum conservation must be applied separately to each independent component (e.g., x and y directions). This is the primary tool used to analyze multi-object collision problems.

Inelastic Collision

An inelastic collision is one in which the colliding bodies stick together after impact, resulting in a single combined mass moving with a common final velocity. In these collisions, while momentum is conserved, kinetic energy is not, and some of it is transformed into other forms such as heat or sound.

Vector Decomposition

Many collision problems occur in a two-dimensional plane and require breaking down velocities into their component parts using trigonometric relationships. This allows the individual momentum contributions in the x and y directions to be analyzed separately, ensuring accurate application of conservation laws.

Kinetic Energy Transformation

In inelastic collisions, a portion of the mechanical energy (kinetic energy) is converted into other forms of energy due to deformation, heat, or sound. Calculating the change in kinetic energy provides insight into the inefficiencies of such collisions and helps distinguish between energy conservation and momentum conservation in collision analyses.

Transcript

00:01 Problem we have given three spheres a sphere like this this is suppose x -axis this is y -axis is sphere a is moving with speed 1 .50 meter per second and sphere b is moving in this direction this angle is 60 degree with the speed 0 .50 meter per second and here another sphere c is moving with vc.

00:33 Suppose this is x component and this is y component here.

00:40 And mass of sphere a is given to us, which is 0 .020 kg.

00:49 And mass of b is given 0 .030 kg.

00:56 And mass of c also given, which is 0 .050 kg.

01:03 This is mass of c, this is mass of b, this is mass of a.

01:09 Now they collide and after collision it is given that they stick together.

01:20 So what must the x and y component of the initial velocity of c? so the final velocity is 0 .50 meter per second in x direction.

01:33 This is final velocity after collision of all three spheres.

01:37 So we have to find the velocity of sphere c.

01:45 So here we will conserve momentum.

01:47 We will write p initial equal to p final in x direction.

01:53 In x direction we will consider momentum first.

01:56 So in x direction due to sphere a momentum is mass of sphere a into velocity of sphere a in x direction plus mass of b velocity of v in x direction plus mass of c and velocity of c in x direction should be equal to m a plus m v plus m c into v final x so here mass of a is 0 .020 and velocity of a in x direction is negative 1 .50.

02:40 Meter per second plus because it is in the left direction towards negative x -axis.

02:46 Mass of v is 0 .030 into v bx will be equal to vb which is 1 .50 into cos 60 and it is in negative.

03:11 So negative sign will be there plus mass of c which is 0 .050 kg into to vcx, this we have to find here and m a plus mb plus mc will be equal to this is 0 .020...

Please give Ace some feedback