6.06a Motion under variable forces and differential equations of motion Quiz: OCR Further Maths, Unit 3

The 20 questions

1. Which expression for acceleration should be used when velocity is given as a function of displacement x?

   • v dx/dt
   • d^2x/dt^2
   • v dv/dx
   • dv/dt

2. Which expression for acceleration is used when velocity is explicitly given as a function of time t?

   • dv/dt
   • d^2t/dx^2
   • v dx/dt
   • v dv/dx

3. A particle of mass 2 kg moves under a force F = 6t. What is its acceleration at t = 3?

   • 3 m s^-2
   • 9 m s^-2
   • 36 m s^-2
   • 18 m s^-2

4. What method solves a differential equation of the form dv/dt = g(t)h(v)?

   • Separation of variables
   • Partial fractions
   • Integrating factor
   • Euler's method

5. What is the integrating factor for the first-order linear differential equation dv/dt + P(t)v = Q(t)?

   • e^(-∫ P dt)
   • ∫ P dt
   • e^(∫ Q dt)
   • e^(∫ P dt)

6. If v dv/dx = 4x and v = 0 at x = 0, what is v in terms of x?

   • 2x
   • 2x^2
   • 4x
   • x^2

7. A particle moves with acceleration a = -kv. What type of differential equation models this motion?

   • Second-order linear
   • Exact quadratic
   • Non-linear homogeneous
   • First-order linear

8. If acceleration a = 3v, which separable differential equation determines displacement x in terms of v?

   • v dv/dx = 3
   • dv/dx = 3v
   • dx/dv = 3v
   • dv/dx = 3

9. A force F = 8 - 2v acts on a 1 kg particle. What is its terminal velocity?

   • 8 m s^-1
   • 2 m s^-1
   • 4 m s^-1
   • 0 m s^-1

10. Integrating dv/dt = 10 - 2v using an integrating factor requires rewriting it in which standard form?

    • dv/dt - 2v = 10
    • dv/dt + 2v = 10
    • 2v dv/dt = 10
    • dv/dt + 10v = 2

11. What is the velocity v(t) if dv/dt = -2v with initial velocity v(0) = 5?

    • 5 e^(2t)
    • 2 e^(-5t)
    • 5 e^(-2t)
    • 5 - 2t

12. For motion under a variable force F(x), what does Newton's second law state?

    • m d^2x/dv^2 = F(x)
    • m dv/dx = F(x)
    • m v dv/dt = F(x)
    • m v dv/dx = F(x)

13. A particle has acceleration a = 6t - 2. If v(0) = 3, what is v(2)?

    • 14 m s^-1
    • 8 m s^-1
    • 10 m s^-1
    • 11 m s^-1

14. What is the integrating factor for the differential equation dv/dt + 3t^2 v = t^2?

    • t^3
    • e^(t^3)
    • e^(3t^2)
    • e^(3t)

15. If a = v^2 and v(0) = 1, what is v in terms of x when x(0) = 0?

    • 1 + x
    • e^(2x)
    • 1/(1-x)
    • e^x

16. What condition defines the maximum velocity of a particle moving in a straight line under a variable force?

    • Force is maximum
    • Velocity is zero
    • Acceleration is zero
    • Displacement is zero

17. If a particle of mass 3 kg has acceleration a = 4x, what is the force acting on it?

    • 12x
    • 4x/3
    • 12x^2
    • 4x

18. Which equation links displacement x and velocity v when acceleration is a function of velocity, f(v)?

    • dx/dv = 1 / f(v)
    • dx/dv = f(v) / v
    • dx/dv = f(v)
    • dx/dv = v / f(v)

19. Solve v dv/dx = 9 given v = 0 at x = 0. What is v at x = 2?

    • 3
    • 18
    • 36
    • 6

20. If dv/dt + v = e^(-t) and v(0) = 0, what is the integrating factor?

    • e^(2t)
    • e^t
    • e^(-t)
    • t

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