CHAPTER X. OBLIQUE IMPACT. 129. An impact which is not direct, as defined in Art. 65, is said to be oblique. 130. When two particles impinge obliquely, the line of action of the stress between the particles set up by the impact is called the line of impact. 131. In order to find the velocity of the particles after impact, the velocity before impact of each particle must be resolved into two resolved parts, one along, the other perpendicular to, the line of impact; Then, since the stress is in the line of impact, it has no effect on either of the resolved parts of the velocities perpendicular to this line. The stress produces its effect on the resolved parts of the velocities in the line of impact exactly as if the other resolved parts did not exist. So that we find the velocities in the line of impact after impact as in Art. 72. Each of these new velocities is then compounded with the resolved part perpendicular to the line of impact (which is unaltered) of the velocity before impact. The resultant velocities thus obtained are the required velocities after impact of the particles respectively. Example. Two spheres A and B, of m lbs. and m' lbs. having velocities u velos and u' velos, making angles a and a' respectively with the line of impact, impinge on one another; find their velocities after impact. The resolved parts of the velocities perpendicular to the line of impact are in velos, u sin a, u sin a', respectively. These are unaltered by the impact. u cos a, u' cos a', respectively. Let the elasticity of the spheres be e. B Then, since the problem is now one of direct impact, we have, by Arts. 68 and 72, if v, o be the velocities after impact in the line of impact mu+ niv=mu cos a + m'u' cos a', and, v-v= = -e (u cos a – u' cos a') (m - eni) u cos a + m' (1+e) u' cosa' whence, m + m' m+m' The resultant velocity of A is therefore V{uo sino a +34) in the u sin a direction making the angle whose tangent is with the line of impact; a similar statement gives the velocity of B. V EXAMPLES. XXXIII. 1. A sphere A of elasticity e impinges with 20/2 velos on an equal sphere B at rest, the line of impact making an angle of 45° with the direction of motion of A; find the velocity of B after impact. 2. A sphere impinges on an equal sphere at rest, find the condition that after impact their velocities may be at right angles. 3. A sphere A impinges on a sphere B of equal mass; their velocities before impact are at right angles and equally inclined to the line of impact and are equal in magnitude; shew that when e=1/3 their velocities after impact are inclined at an angle 60'. 4. A glass ball of elasticity e when moving horizontally with velocity u velos, is struck by an equal ball moving vertically with u velos, so that the line of impact is vertical; find their velocities after impact. 5. A bird weighing 1 lb. moving horizontally with 80 velos is struck by a bullet weighing i oz. moving vertically with a velocity of 1020 velos; find the subsequent velocity of the bird supposing the bullet to lodge in the bird ; and supposing it killed by the shot when at a height of 100 ft., find when it will fall to the ground, neglecting the resistance of the air. 6. A bird weighing 2 lbs. moving horizontally with 30 velos is struck by a bullet of 1 oz. moving horizontally, but at right angles to the path of the bird, with a velocity 1320 velos; find the velocity after impact supposing the bullet to lodge in the bird; and supposing the bird killed when at a height of 128 ft., find how long it will be before it strikes the ground. 132. When an elastic ball impinges obliquely on a plane fixed to the ground, the problem is treated as in Art. 73. Examples. A sphere of elasticity e is projected with velocity i at an angle a to the horizon from a point in a smooth horizontal plane, against which it impinges and rebounds; investigate the motion. The resolved parts of the velocity of projection are (in velos) u sin a, vertically; u cos a, horizontally. The particle first describes a parabola. On reaching the horizontal plane again the particle has velocity - usin a, vertically; u cos a horizontally. The impact does not alter the horizontal velocity; the vertical velocity after impact is eu sin a. The particle now describes another parabola the velocity of projection being u cos a velos horizontally, eu sin a velos vertically. And SO on. Thus, after the nth impact the vertical velocity is en u sin a; the horizontal velocity is u cos a. The range of any rebound is (in feet) 2 (vertical velocity x horizontal velocity), + e The distance from the point of projection at which the sphere will strike the plane after the nth rebound is 2uo sin a cos a {itetés + ten}; the limit of which, when n is increased without limit, is 2uo sin a cos a & (I -e) The meaning of this is, that after reaching this distance from the point of projection, the sphere will cease to rebound and will slide with velocity u cos a. 133. The following proposition is important. PROP. When two masses of m lbs. and m' lbs. respectively of elasticity e impinge directly, if v velos and v' velos be their velocities after impact then my® + m'v' is less than mu? + m'u'. We have, mzi + m'v' = mu + m'u and, V-U e (u – u'). Therefore (mv + m'v') = (mu + m'u')? and, mm' (v – v') = mm'e® (u – u'). Whence, by addition (m + m') (mve + m'v*)=(m + m') (mu? + m'u®) - mm' (1 - 0) (u – u), or, mvé + m'v' = mu? + m'um (I –ư°)(x – 1)”. Now e is never greater than 1; so that i e cannot be negative; hence mvé + m'u is always less than mu? + m'u* ; except when e=1, and in that case the two expressions are equal. This result may be expressed (See Chapter XV.) thus; The kinetic energy of two masses is diminished by direct impact, except when the elasticity of the masses is perfect. We leave as an exercise for the student the proof of the proposition that if v velos and v' velos be the velocities after oblique impact of masses m lbs. and m' lbs. then mve + m'y' is less than mu® + m'u“. m m' EXAMPLES. XXXIV. 1. A glass ball of elasticity įrds is projected with 40 velos at an angle of projection whose sine is ß, from a point on a horizontal pavement; find its range after one rebound. 2. A glass ball of elasticity } is projected at an angle whose sine is 13 with 136 velos from a point in a horizontal plane; find how far it will go before it ceases to rebound. 3. A particle of elasticity is projected with velocity 100 velos at an angle 60° to the horizon from a point in an inclined plane making an angie 30° with the horizon; find the velocity of the particle after one rek ound from the plane. 4. A particle of elasticity e is projected with u velos at an angle a to the worizon and after striking a fixed vertical wall at a horizontal distance h ft. returns to the point of projection ; prove that hg (1 +e)=2uo e sin a cos a. 5. Two particles of elasticity e are projected in exactly opposite directior s from a point between two fixed vertical parallel planes and distant a ft. and b ft. from them respectively; shew that they will not meet again unless the directions of projection are perpendicular to the planes. 6. A particle of elasticity e is projected from a point half-way between two fixed parallel vertical walls 2a ft. apart, in a given direction, and after three rebounds returns to the point of projection; find the velocity of projection. Note. When a sphere strikes a plane the angle its velocity makes with the normal to the plane is called the angle of incidence; the angle which its velocity after impact makes with this normal is called the angle of reflexion. 7. A sphere of elasticity & impinges on a plane ; find the angle of incidence that its direction after impact may be at right angles with its direction before impact. 8. Prove that when a perfectly elastic sphere impinges on a smooth plane the angles of incidence and reflexion are equal. 9. A perfectly elastic ball is projected on a smooth rectangular billiard table in a direction parallel to one of its diagonals; find the condition that after impinging on each of the four sides the ball will return to the point of projection. 10. If a be the angle of incidence and B the angle of reflexion of a sphere of elasticity e on a smooth plane, prove that.cot ßre cot a. |