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

Did this hint help?