Paper 1

Question 1

Hint 1: Use equations of uniform motion, find acceleration, find final velocity,

Hint 2: Work out further distance, stopping distance and total distance

Hint 3: and here is a video of the solution:

Question 2

2a) Hint 1: Use correct form of partial fractions: 3 distinct linear factors and process

2b) Hint 2: Integrate using partial fractions

2b) Hint 3: Recognise standard log integrals (take care of negatives)

2b) Hint 4: Use rules of logs to rewrite answer in required format

Hint 5: and here is a video of the solution:

Question 3

Hint 1: Use Newton's second law with frictional force to work out deceleration

Hint 2: Use equations of uniform motion to calculate velocity

Hint 3: Use conservation of momentum given that the bodies coalesce to find velocity

Hint 4: and here is a video of the solution:

Question 4

Hint 1: Differentiate using standard derivatives and chain rule

Hint 2: evaluate expression

Hint 3: and here is a video of the solution:

Question 5

Hint 1: Draw a diagram and resolve horizontally and vertically using F = mrw²

Hint 2: Find tan(θ) by getting ratio of horizontal and vertical

Hint 3: Use information about triangle side lengths to get expression for tan(θ), equate and rearrange

Hint 4: and here is a video of the solution:

Question 6

Hint 1: Express volume as integral by π times the integral of y² with respect to x

Hint 2: Integrate and evaluate

Hint 3: and here is a video of the solution:

Question 7

7a) Hint 1: Find ω

7a) Hint 2: Know x = a sin(ωt + α), and work out value of α when t=0 when at centre of oscillation (α = 0)

7a) Hint 3: Substitute and solve for first two times

7b) Hint 4: Know v is derivative of above, work out and substitute

7b) Hint 5: Interpret the sign of velocity information

Hint 6: and here is a video of the solution:

Question 8

Hint 1: Work out dx/dt and dy/dt

Hint 2: Evaluate for t = 3

Hint 3: Know how the speed relates to the magnitude of the above (using pythagoras' theorem)

Hint 4: and here is a video of the solution:

Question 9

9a) Hint 1: Draw a diagram and derive basic results for horizontal and vertical for acceleration, velocity and displacement

9a) Hint 2: Know that at range, y = 0 and find expression for t

9a) Hint 3: Substitute value of t into expression for x and rearrange as required

9b) Hint 4: Substitute both angles into range formula

9b) Hint 5: Note one is R, the other R+5

9b) Hint 6: Set up from above and solve for v

9b) Hint 7: Obtain range given angle is 35°

9b) Hint 8: Find the time of flight

9b) Hint 9: Find extra distance given tail wind and add to range above

Hint 10: and here is a video of the solution:

Question 10

10a) Hint 1: Derive expressions for mass and centre of mass of the circular lamina and the circular hole

10a) Hint 2: Derive expressions for mass and centre of mass of the semi-circular hole

10a) Hint 3: Take moments horizontally and solve to find x

10a) Hint 4: Take moments vertically and solve to find y

10b) Hint 5: Draw a diagram and interpret rotation

Hint 6: and here is a video of the solution:

Question 11

11a) Hint 1: Calculate displacement of A and B

11a) Hint 2: Calculate velocity given uniform motion

11b) Hint 3: Express displacement of A and B as functions of time

11b) Hint 4: Know that when they collide, i and j components are equal for the same value of t

11b) Hint 5: Find the displacement given the value of t above.

Hint 6: and here is a video of the solution:

Question 12

12a) Hint 1: Use Newton's second law parallel to wire

12a) Hint 2: Resolve perpendicular to cable and get expression for acceleration

12a) Hint 3: Use equations of uniform motion to calculate velocity

12b) Hint 4: Find total initial energy and final energy

12b) Hint 5: Use conservation of energy to form equation

12b) Hint 6: Substitute values and calculate angle

Hint 7: and here is a video of the solution:

Question 13

Hint 1: Find an expression for du

Hint 2: Evaluate new limits

Hint 3: Make the substitution

Hint 4: Simplify and do the integration and evaluate

Hint 5: and here is a video of the solution:

Question 14

Hint 1: Model elastic potential energy (PE) in rope

Hint 2: Equate PE and elastic PE at lowest point

Hint 3: Set up quadratic equation and solve

Hint 4: Interpret and select appropriate solution and complete

Hint 5: and here is a video of the solution:

Question 15

15a) Hint 1: Recognise second order homogeneous differential equation and set up auxiliary equation

15a) Hint 2: Solve quadratic equation to give general solution

15a) Hint 3: Substitute initial conditions for x and dx/dt to find constants

15b) Hint 4: Substitute for t and work out distance moved (remember to interpret x in the context)

Hint 5: and here is a video of the solution:

Question 16

16a) Hint 1: Sketch graph showing speed against time for both runners and annotate

16b) Hint 2: Find the displacement for P

16b) Hint 3: Find the displacement for Q

16b) Hint 4: Interpret what the displacements mean (diagram is helpful) and calculate distance

Hint 5: and here is a video of the solution:

Question 17

Hint 1: Use Newton's second law , choose appropriate expression for a

Hint 2: Choose a = dv/dt

Hint 3: Know Impulse is change in momentum

Hint 4: Solve differential equation by method of separation of variables, and integrate

Hint 5: use initial conditions t = 0 and v = I/m

Hint 6: Manipulate to complete proof

Hint 7: and here is a video of the solution:

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