A sphere of mass m 1 impinges with a velocity v 0 on a sphere of mass m 2 that is at rest, with m 1 > m 2 . The collision is absolutely elastic but not central. By what maximal angle will the impinging sphere be deflected?

Let u 1 and u 2 be the final velocities of the impinging sphere and the one that was at rest prior to collision, respectively, and θ is the angle between u 1 and v 0 . The equations that express the laws of conservation of energy and momentum (for each projection) have the following form. m 1 v 0 2 2 = m 1 u 1 2 2 + m 2 u 2 2 2 (Conservation of Energy) and m 1 v 0 = m 1 u 1 cos θ + m 2 u 2 cos φ (Conservation of Momentum along the line of initial momentum) and m 1 u 1 sin θ + m 2 u 2 sin φ = 0 (Conservation of Momentum along the line perpendicular to that of initial momentum). If m 1 , m 2 , and v 0 are fixed, then u 1 , u 2 , θ , and φ are linked through three equations. For this reason two of the four variables can be excluded and the variable θ can be expressed in terms of the third remaining variable, say, u 1 . Taking m 1 u cos θ to the left-hand side of equation. On squaring the result, and adding the two squares, we get m 1 2 v 0 2 - 2 u 1 u 2 cos θ + u 1 2 = m 2 2 u 2 2 . Replacing u 2 with its value obtained and carrying out the necessary transformations, we arrive at a quadratic equation for u 1 , namely, u 1 2 - 2 m 1 m 1 + m 2 v 0 cos θ × u 1 + m 1 - m 2 m 1 + m 2 v 0 2 = 0 , (1.33.4) whose solution has the form u 1 = m 1 m 1 + m 2 cos θ ± m 2 m 1 2 - sin 2 θ v 0 . This equation shows that the maximal angle θ is determined by the condition sin θ m = m 2 m 1 . For values of θ smaller than θ m two cases are possible, since two distinct values of u 1 corresponding to one value of θ . For example, for m 1 m 2 = 3 and sin θ = 0 . 2 , the velocity u 1 may have two values, 0 . 93 v 0 and 0 . 53 v 0 . The first collision is commonly known as soft, while the second is commonly known as hard. The extreme case of soft collision is the grazing collision (or even the case where one sphere misses the other), while the extreme case of hard collision is the head-on collision, after which the velocity of the impinging sphere becomes u 1 = m 1 - m 2 m 1 + m 2 v 0 . Condition can be obtained in another manner as well. For instance, if we express cos θ via, namely, cos θ = 1 2 m 1 m 1 + m 2 u 1 v 0 + m 1 - m 2 v 0 u 1 , For instance, if we express cos θ , namely, cos θ = 1 2 m 1 m 1 + m 2 u 1 v 0 + m 1 - m 2 v 0 u 1 , and nullify the derivative of cos θ with respect to u 1 , we can find the minimal value of cos θ or the maximal value of sin θ . The motion of the impinging sphere can also be considered using the system of coordinates linked with the center of mass of the two spheres. If in the laboratory system the coordinates of the spheres are x 1 and x 2 , then the coordinate of the center of mass is x = m 1 x 1 + m 2 x 2 m 1 + m 2 , while the velocity of the center of mass is v c = m 1 m 1 + m 2 v 0 . Correspondingly, the velocity of the impinging sphere in this system prior to the collision is v 0 ' = v 0 - v c = v 0 . As a result of the collision, the vector v 0 ' retains its length but turns through a certain angle depending on the distance between the centre of the second sphere and (a) (b) (c) the direction of flight of the impinging sphere prior to the collision. The velocity u 1 is equal to the sum of v 0 ' and v c . The momentum vectors of both spheres are shown in the figure for three cases: soft collision (Figure (a)) and hard collision (Figure (b)) for m 1 m 2 = 3 and sin θ = 0 . 2 and the case with sin θ = m 2 m 1 = 1 3 (Figure (c)). The velocity of the impinging sphere after the collision is u 1 = m 1 v 0 m 1 + m 2 cos θ = 0 . 707 v 0 . The above-discussed problem is important for the theory of atomic collisions. For instance, if a potassium ion impinges on a helium atom m 1 m 2 = 10 , as a result of an elastic collision the ion may be deflected by an angle no greater than 5 . 7 ° .

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Two spheres of equal mass collide, with the collision being absolutely elastic but not central. Prove that in this case the angle between the velocities after collision must be $90 °$ .

Important Questions on Fundamentals of Mechanics

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.33

A sphere of mass

u_{2}

impinges with a velocity

u_{1}

on a sphere of mass

u_{2}

that is at rest, with

90 °

. The collision is absolutely elastic but not central. By what maximal angle will the impinging sphere be deflected?

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.34

Two spheres of equal mass are moving at right angles with velocities that are equal in magnitude. At the moment of collision the velocity vector of sphere $1$ is directed along the straight line connecting the centers of the spheres. The collision is absolutely elastic. Plot the velocity vectors before and after collision in different coordinate systems: ( $1$ ) in the laboratory system (in this system the velocities of the spheres are those specified above), ( $2$ ) in the coordinate system connected with the center of mass of the two spheres, and ( $3$ ) and ( $4$ ) in the coordinate systems linked to each of the spheres.

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.35

The centres of the spheres $1, 2,$ and $3$ lie on a single straight line. Sphere $1$ is moving with an (initial) velocity $v_{1}$ directed along this line and hits sphere $2$ . Sphere $2$ , acquiring after collision a velocity $v_{2}$ , hits sphere $3$ . Both collisions are absolutely elastic. What must be the mass of sphere $2$ for the sphere $3$ to acquire maximum velocity (the masses $m_{1}$ and $m_{3}$ of spheres $1$ and $3$ are known)?

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.36

A sphere of mass

m_{1}

moving with a velocity

v_{0}

hits a sphere of mass

m_{2}

that is at rest. The collision is absolutely elastic and central. The velocities of the spheres after collision are

u_{1}

and

u_{2}

respectively. What are the mass ratios for the following values of velocities:

u_{1} = 0, u_{1} < 0

, and

u_{1} > 0

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.37

A device often used to illustrate the laws of uniformly accelerated motion is the Atwood machine. The machine consists of two loads of mass $m_{1}$ and $m_{2}$ attached to the ends of a limp but inextensible string. The string runs over a pulley. The acceleration with which the loads move is $w = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} g$ , whereas the angular acceleration of the pulley is ignored. Is the last assumption true for exact calculations?

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.38

Strings are wound around a shaft and a sheave of equal mass, and a load is attached to the end of each string (the loads have equal mass). Which of the two loads will descend with a greater acceleration and which of the rotating objects, the shaft or the sheave, has a greater angular acceleration?

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.39

A vacuum cleaner standing on the floor turns through a small angle when switched on and then stops. Why does this happen?

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A Collection of Questions and problems in Physics>Chapter 1 - Fundamentals of Mechanics>Exercise>Q 1.4

A number of types of helicopters, among which are the Soviet-made "Mi" helicopters and the Westland Whirlwinds designed for use by Queen Elizabeth $II$ , utilize one main rotor and a small vertical tail rotor. What is the function of this second rotor?

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Two spheres of equal mass collide, with the collision being absolutely elastic but not central. Prove that in this case the angle between the velocities after collision must be 90°.

Important Questions on Fundamentals of Mechanics

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Two spheres of equal mass collide, with the collision being absolutely elastic but not central. Prove that in this case the angle between the velocities after collision must be $90 °$ .