Concept explainers
In the scattering experiment of Prob. 14.26, it is known that the alpha particle is projected from A0(300, 0, 300) and that it collides with the oxygen nucleus C at Q(240, 200, 100), where all coordinates are expressed in millimeters. Determine the coordinates of point B0 where the original path of nucleus B intersects the zx plane. (Hint: Express that the angular momentum of the three particles about Q is conserved.)
14.26 In a scattering experiment, an alpha particle A is projected with the velocity u0 = −(600 m/s)i + (750 m/s)j − (800 m/s)k into a stream of oxygen nuclei moving with a common velocity v0 = (600 m/s)j. After colliding successively with the nuclei B and C, particle A is observed to move along the path defined by the points A1 (280, 240, 120) and A2 (360, 320, 160), while nuclei B and C are observed to move along paths defined, respectively, by B1 (147, 220, 130) and B2 (114, 290, 120), and by C1 (240, 232, 90) and C2 (240, 280, 75). All paths are along straight lines and all coordinates are expressed in millimeters. Knowing that the mass of an oxygen nucleus is four times that of an alpha particle, determine the speed of each of the three particles after the collisions.
Fig. P14.26
The coordinates of point
Answer to Problem 14.48P
The coordinates of point
Explanation of Solution
Given information:
The velocity of the alpha particle A is
The common velocity of oxygen nuclei is
The alpha particle A projected from
The position of the alpha particle A is
The position of the nuclei B is
The position of the nuclei C is
The mass of an oxygen nucleus is four times that of an alpha particle.
The alpha particle collides with the oxygen nucleus C at
Calculation:
Provide the positions of each point in vector form as shown below.
Sketch the scattering of the alpha and nuclei particles as shown in Figure 1.
Refer to Figure 1.
Calculate the position vectors as shown below.
Calculate
Substitute
Calculate
Substitute
Calculate
Substitute
Calculate
Substitute
Calculate
Substitute
Calculate
Calculate the unit vector
Substitute
Calculate the unit vectors
Substitute
Calculate the unit vectors
Substitute
Provide the velocity vectors after the collisions as shown below.
Apply the conservation of momentum as shown below.
Substitute
Substitute
Equating the components of
Solve the Equations to get the speed of the particles.
Calculate the velocity vector
Substitute
Apply the conservation of momentum about Q as shown below.
Substitute
Equating the components of
Refer to Figure 1.
Calculate the coordinates of
Coordinate of along x direction.
Substitute
Coordinate of along y direction.
Coordinate of along z direction.
Substitute
Therefore, the coordinates of point
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Chapter 14 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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