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Paper 1
Question 1
Hint 1: Know that impulse is change of momentum
Question 2
2a) Hint 1: Differentiate using the product rule and chain rule and evaluate
2b) Hint 2: Differentiate using the quotient rule and chain rule and simplify including factorising
Question 3
Hint 1: Know that displacement is the integral of velocity.
Hint 2: Know that range is the magnitude of displacement
Question 4
Hint 1: Use results for maximum velocity and acceleration
Hint 2: Know that displacement is given by x = a sin(wt + α), velocity is derivative of that
Hint 3: Interpret negative sign
Question 5
Hint 1: Solve second order differential equation: solve auxiliary equation, using boundary conditions
Question 6
6a) Hint 1: Know moments are product of force and distance,
6a) Hint 2: Work out resultant and state direction
6b) Hint 3: Equate clockwise and anticlockwise moments
Question 7
Hint 1: Use standard integration results including chain rule
Hint 2: simplify including factorisation
Question 8
Hint 1: Know velocity is integral of acceleration using boundary conditions
Hint 2: For integration by parts choose u=2t etc
Question 9
Hint 1: Resolve parallel and perpendicular to the plane
Hint 2: Take the ratio of expressions for Fcos(θ) and Fsin(θ) to get tan(θ)
Hint 3: Substitute back into expression for F
Question 10
Hint 1: Differentiate implicitly
Hint 2: Make dy/dx the subject and evaluate
Question 11
Hint 1: Draw a diagram and establish equation of motion
Hint 2: Choose a=vdv/dx and solve differential equation
Question 12
12a) Hint 1: Draw a diagram, min speed without slipping - which direction will friction be
Hint 2: Resolve horizontally and vertically
Hint 3: Horizontal force towards centre = mv²/r
Hint 4: Equate expressions for R, and rearrange
12b) Hint 5: Interpret whether there will be slipping - consider friction down the slope, continue as part (a)
12b) Hint 6: Interpret result
12c) Hint 7: Choose a physical reason
Question 13
13a) Hint 1: Draw a diagram and establish equation of motion ( equilibrium or not?)
13a) Hint 2: Resolve parallel and perpendicular to the plane
13a) Hint 3: Use equations of uniform motion
13b) Hint 4: Work energy principle , work done is force times distance, rearrange
Question 14
14a) Hint 1: Draw a diagram - vertical circles
14a) Hint 2: Consider conservation of energy
14b) Hint 3: Condition to exit pipe: V ≥ 0 when θ = 180°
14b) Hint 4: One physical assumption
Question 15
15a) Hint 1: Draw a diagram - and establish basic horizontal and vertical equations for acceleration, velocity and displacement
15a) Hint 2: Max height when vertical velocity is zero and interpret
15b) Hint 3: Get expression for range and use previous result for sin(θ) and use trig identity to get value for cos θ
15c) Hint 4: Range needs to be a real number - implication for discriminant
Question 16
16a) Hint 1: Draw a diagram, use displacement diagram (triangle)
16a) Hint 2: Sine rule and cosine rule
16b) Hint 3: Update diagram, continue with similar approach to part (a)
Question 17
Hint 1: Integrate by inspection
Hint 2: Recognise expression for velocity as original expression that is to be integrated.
Hint 3: Remember what 'at rest' means about velocity
Hint 4: Reason that each of the two functions, the product of which is the velocity, cannot be zero