M1, C5 - Elastic Collisions in 2 Dimensions

Question 1

What is the equation for e when the impact is oblique

Answer 1

e = vsinβ / usinα

Question 2

What is the equation of motion parallel to the fixed surface

Answer 2

Velocity stays constant
ucosα = vcosβ

Question 3

What is the equation of motion perpendicular to the fixed surface

Answer 3

e * usinα = vsinβ

Question 4

What happens to the velocity parallel to the wall

Answer 4

Stays constant

Question 5

vcos(β) = u/2, vsin(β) = (root(3) * u) / 8
Without finding β, how do you find v

Answer 5

(vsin(β))^2 + (vcos(β))^2 = ((root(3) * u) / 8)^2 + (u/2)^2
v^2 * (sin^2(β) + cos^2(β)) = ((root(3) * u) / 8)^2 + (u/2)^2
v^2 = ((3u^2) / 64) + (u^2)/4
v^2 = 19u^2 / 64
v = (root(19) * u) / 8

Question 6

What is the equation for the angle of deflection

Answer 6

α + β
cos^-1 ((a.b) / (lallbl))

Question 7

What is the equation for the angle of deflection is the velocities are vectors

Answer 7

cos^-1 ((a.b) / (lallbl))
Where a and b are the given velocities

Question 8

A particle travelling at -6i - 4j collides with a vertical plane with coefficient of restitution 1/3. What is the velocity after the impact

Answer 8

Perpendicular component:
-6 * 1/3 = -2i
Parallel component
-4j = -4j
Answer = -2i - 4j

Question 9

How do you find a unit vector of an impulse

Answer 9

Divide the impulse by it’s length

Question 10

How do you find the perpendicular component of u when velocities are given as vectors

Answer 10

I . u = k
ans = k * I
where I is the impulse

Question 11

How do you find the perpendicular component of v when velocities are given as vectors

Answer 11

I . v = P
ans = P * I
where I is the impulse

Question 12

How do you find the parallel component of u when velocities are given as vectors

Answer 12

(k . u) * k = ans
where k is a unit vector perpendicular to the impulse (I)

Question 13

How do you find the parallel component of v when velocities are given as vectors

Answer 13

(k . v) * k = ans
where k is a unit vector perpendicular to the impulse (I)

Question 14

A sphere of mass m moves with velocity 5i - 2j and collides with a smooth wall rebounding with velocity 2i + 2j, how do you find a) the unit vector in the direction of the impulse received by the sphere
b) the coefficient of restitution

Answer 14

a) I = mv - mu
I = m((2i + 2j) - (5i - 2j))
I = m(-3i + 4j)
Unit vector of I = 1/5 * (-3i + 4j)
b) Unit vector of I . u = -23/5
-23/5 * (-3/5, 4/5) = 69/25 i - 92 / 25j
Unit vector of I . v = 2/5
2/5 * (-3/5, 4/5) = -6/25 i + 8/25 j
e = (2/5) / (23/5) = 2/23

Question 15

When given velocities as vectors, how can you find e without working out the perpendicular components of velocity

Answer 15

Dot product of unit vector I and v / dot product of unit vector I and u
Where I is impulse

Question 16

What is the notation for a unit vector (u)

Answer 16

u with _ underneath and ^ above

Question 17

What happens to velocities perpendicular to the line of centres

Answer 17

Remains unchanged

Question 18

What components of velocity are acted on by Newton’s law of restitution with smooth spheres

Answer 18

Velocities parallel to the line of centres

Question 19

What velocities do you use for your conservation of linear momentum, coefficient of restitution, and impulse calculations with smooth sphere collisions

Answer 19

Parallel velocities to the line of centres

Question 20

A 2kg sphere (A) moving at 6ms^-1 collides obliquely with a 4kg stationary sphere (B) and the velocity of A make an angle of 60 degrees with the lines of centres. e = 1/4. Find the magnitudes and direction of A and B after the impact

Answer 20

Before:
A: Perpendicular 6sin60, Parallel 6cos60
B: Perpendicular 0, Parallel 0
After:
A: Perpendicular 6sin60
B: Perpendicular 0
CLM:
2*6cos(60) = 2v + 4w = 6
e:
6cos60 = 3
1/4 = (w-v) / (3) –> 4w - 4v = 3
v = 1/2, w = 5/4
A: Parallel 1/2
B: Parallel 5/4
Magnitude and Direction of total A and B velocities:
A = root(109) / 2 m/s, 84.5 degrees
B = 5/4 m/s along line of centres (0 degrees)

Question 21

What do you do if given velocities as vectors (or otherwise) for smooth sphere oblique impacts and the line of centres are not parallel (horizontal) or perpendicular (vertical)

Answer 21

Form vector equations (impulse, CLM, e) and solve (no need for diagram)

Question 22

How do you find a unit vector parallel to the lines of centres of spheres at the instant of a collision

Answer 22

The impulse will be parallel to the line of centres
Consider sphere A or B and find the impulse
Find unit vector of impulse for answer

Question 23

What is the unit vector of the impulse (6mi - 6mj)

Answer 23

1 / magnitude * vector
1 / 6mroot(2) * (6mi - 6mj)
ans = (1/root(2)) * (i - j)

Question 24

What diagrams should you draw for oblique impacts of smooth sphere questions

Answer 24

Before and after diagrams