Solving Problems in Elastic Collisions
Solving problems in elastic collisions involving oblique impacts of smooth spheres on fixed surfaces. Understand changes in velocity and momentum during collisions, why angles change, and formulas related to the horizontal velocity component. Explore scenarios of normal and oblique collisions, coefficient of restitution, and loss of kinetic energy due to impacts.
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Prior Knowledge Check 2) A particle ? of mass 1.5kg lies at rest on a smooth horizontal table. A second particle ? of mass 0.5g is projected along the table with velocity 5ms-1 and collides directly with ?. If the collision reduces the speed of ? to 2ms-1, without changing its direction, find: a) The speed of ? after the collision b) The loss of kinetic energy due to the impact 1) A small smooth sphere of mass 0.25kg is moving on a smooth horizontal table with a speed of 8ms-1 when it collides normally with a fixed smooth wall. It rebounds with a speed of 6ms-1. Find the coefficient of restitution between the sphere and the wall. 0.75 1ms-1 4.5J lost
Elastic Collisions in two dimensions ? You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface ? ? ? ? An oblique impact is one where an object does not strike a surface normally (ie perpendicular to) When a smooth sphere collides with a smooth surface and bounces away, the velocity will change, and hence the momentum also changes The impulse on the sphere will act perpendicular to the surface, through the centre of the sphere Note that the sphere will not necessarily bounce away at the same angle with which it hit the surface why not? 5A
Elastic Collisions in two dimensions You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface Why does the angle change? ? ? ? ????? ? ????? ? ? ????? ????? If a sphere collides with a surface normally (perpendicular to), then the coefficient of restitution will mean it bounces away at a lower velocity When the collision is oblique, the velocity is split into two components, parallel to the plane, and perpendicular to it The parallel component s actual velocity will not change after the impact, since the surfaces are smooth The perpendicular component will however be reduced by the coefficient of restitution, hence the angle changes 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface ? ????? ? ????? ? ? ????? ????? This relationship leads to several formulae The horizontal component of the velocity does not change The speed of separation is the speed of approach, multiplied by the coefficient of restitution You can divide the second equation by the first ????? ?????=?????? ????? = ????? ????? = ?????? ????? ???? = ????? 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface Wall Plane A smooth sphere ? is moving on a smooth horizontal plane with speed ? when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of ? makes an angle of 60 with the wall. The coefficient of restitution between ? and the wall is 1 Sphere Since the ball is moving in a horizontal plane, we will be ignoring gravity 4. Find: Remember this when we do a sketch it is essentially an overhead view of the situation! a) The speed of ? immediately after the collision b) The angle of deflection of ? 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions ????? = ????? We know ? Calculate You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 60 ????? = ????60 ? ????? =1 2? A smooth sphere ? is moving on a smooth horizontal plane with speed ? when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of ? makes an angle of 60 with the wall. The coefficient of restitution between ? and the wall is 1 ? ? ????? = ?????? We know ? and ? 1 4 ????? = ????60 4. Find: Calculate 3 a) The speed of ? immediately after the collision b) The angle of deflection of ? ????? = 8? We can eliminate ? by using one of the trigonometrical identities. We will need to square both equations first You can form simultaneous equations to help solve part a ????? =1 3 2? ????? = 8? Square Square ?2???2? =1 3 64?2 4?2 ?2???2? = 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 60 ? A smooth sphere ? is moving on a smooth horizontal plane with speed ? when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of ? makes an angle of 60 with the wall. The coefficient of restitution between ? and the wall is 1 ? ? ?2???2? =1 3 64?2 4?2 Add both equations together! ?2???2? = 4. Find: a) The speed of ? immediately after the collision b) The angle of deflection of ? ?2???2? + ?2???2? =1 3 64?2 4?2+ Factorise left side ?2???2? + ???2? =19 You can form simultaneous equations to help solve part a 64?2 The bracketed part = 1 ?2=19 64?2 Square root both sides 19 64? ? = 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions The angle of deflection of ? is the angle which the direction of motion has rotated through You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 60 ? Check where S would go if the wall were not present A smooth sphere ? is moving on a smooth horizontal plane with speed ? when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of ? makes an angle of 60 with the wall. The coefficient of restitution between ? and the wall is 1 ? ? ?? ? You can see from this that the angle of deflection will be the sum of angles ? and ? 4. Find: So to answer this question we need to find angle ?, and add on 60 a) The speed of ? immediately after the collision b) The angle of deflection of ? 19 64? ? = 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface ???? = ????? Sub in values 60 ? 1 4 ???? = ???60 A smooth sphere ? is moving on a smooth horizontal plane with speed ? when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of ? makes an angle of 60 with the wall. The coefficient of restitution between ? and the wall is 1 Calculate ? 3 ? ???? = 4 Calculate ? = 23.41 4. Find: Add on 60 to find the angle of deflection a) The speed of ? immediately after the collision b) The angle of deflection of ? ????? ?? ?????????? = 83.41 19 64? ? = 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions ???? = 2 You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 2 5?? 1 ???? = 5 ? 1 ? ???? = A small smooth ball is falling vertically. The ball strikes a smooth plane which is inclined at an angle ?to the horizontal, where ???? =1 striking the plane, the ball has speed 5?? 1. The coefficient of restitution between the ball and the plane is 1 the speed of the ball immediately after the impact. 5 ? ? ? 2. Immediately before If the angle of approach is ?, the non-90 angle in the triangle will also be ? ???? = ????? 3. Find Sub in values 1 3 ???? = (2) Simplify ???? =2 We can use trig ratios to find an expression for ???? 3 2 ???? =1 5 ???? = 1 ? 5 1 2 ? ???? = 2 ???? = 2 5 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions ???? = 2 You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 2 5?? 1 ???? = 5 ? 1 ? ???? = A small smooth ball is falling vertically. The ball strikes a smooth plane which is inclined at an angle ?to the horizontal, where ???? =1 striking the plane, the ball has speed 5?? 1. The coefficient of restitution between the ball and the plane is 1 the speed of the ball immediately after the impact. 5 ? ? ? ???? =2 2. Immediately before 3 3. Find ????? = ????? Sub in values 3 1 ? = (5) 13 5 We can use trig ratios to find other expressions for ? Rearrange 5 13 3 ? = 5 2 ???? =2 Simplify ???? = 13 3 2 13 65 3 ?? 1 ? = ? 3 3 ???? = 13 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions You can split the movement into the ? and ? components 6? You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface Since the surfaces are smooth, the velocity parallel to the wall will remain constant 4? ? 6? 4? ?? 1 A small smooth ball of mass 2kg is moving in the ?? plane and collides with a smooth fixed vertical wall which contains the ? ????. The velocity of the ball just before impact is 6? 4? ?? 1. The coefficient of restitution between the ball and the wall is 1 4? ?? ? ?? + ?? ?? 1 Only the velocity perpendicular to the wall will be affected by the coefficient of restitution ?? 2? 3. Find: ? =? a) The velocity of the ball immediately after the impact The kinetic energy lost as a result of the impact The angle of deflection of the ball ? Sub in values. The value of ? is taken as positive since it is the speed of approach (not velocity) b) 1 3=? c) 6 Calculate 2 = ? 5A
????? = ????? ????? = ?????? ???? = ????? Elastic Collisions in two dimensions We need to know the speed of the particle before and after, which is given by ? 6? You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface 4? ? 6? 4? ?? 1 A small smooth ball of mass 2kg is moving in the ?? plane and collides with a smooth fixed vertical wall which contains the ? ????. The velocity of the ball just before impact is 6? 4? ?? 1. The coefficient of restitution between the ball and the wall is 1 62+ 42 ? = 4? ? ?? + ?? ?? 1 ? = 52 2? 22+ 42 ? = 3. Find: ? = 20 a) The velocity of the ball immediately after the impact The kinetic energy lost as a result of the impact The angle of deflection of the ball ? ???? ?? ?? =1 2??2 1 2??2 b) Sub in values ? ???? ?? ?? =1 2 1 2 c) 2(2) 52 2(2) 20 Calculate ? ???? ?? ?? = 32? So 32J of energy has been lost due to the impact 5A
?.? ? ? ????? = ????? ????? = ?????? ???? = ????? ???? = Elastic Collisions in two dimensions As seen earlier, the angle of deflection is the angle which the direction of motion has turned though You need to be able to solve problems involving the oblique impact of a smooth sphere on a smooth fixed surface ? 6? 4? ?? 1 A small smooth ball of mass 2kg is moving in the ?? plane and collides with a smooth fixed vertical wall which contains the ? ????. The velocity of the ball just before impact is 6? 4? ?? 1. The coefficient of restitution between the ball and the wall is 1 You can calculate these angles separately, or you can use the scalar product (the angle between two vectors) ? ? 2? 4? ?? 1 3. Find: ?.? ? ? ???? = Sub in values a) The velocity of the ball immediately after the impact The kinetic energy lost as a result of the impact The angle of deflection of the ball 6 4 2 . 4 ???? = b) Write the dot product as a calculation 62+ 42 22+ 42 c) 6 2 + ( 4)(4) ???? = 62+ 42 22+ 42 Calculate ???? = 0.124 Inverse cos ? = 82.9 5A
