Further kinematics Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Further kinematics questions. See exactly how to solve problems on velocity, differentiation, acceleration, substitution.

velocitydifferentiationaccelerationsubstitutionspeedmagnitude
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A particle moves in a plane so that its position vector at time tt seconds is r=t2i+3tj\mathbf{r} = t^{2}\mathbf{i}+3 t\mathbf{j} metres. Find its velocity vector v\mathbf{v} as a function of tt.

Worked solution

  1. Write down the position vector

    r=t2i+3tj\mathbf{r}=t^{2}\mathbf{i}+3 t\mathbf{j}

    The motion is described by this position vector; velocity is its rate of change.

  2. Differentiate each component with respect to time

    drdt=2ti+3j\frac{d\mathbf{r}}{dt}=2 t\mathbf{i}+3\mathbf{j}

    Differentiating each component gives the velocity vector.

  3. State the velocity vector

    v=2ti+3j\mathbf{v}=2 t\mathbf{i}+3\mathbf{j}

    This is the velocity as a function of time.

Answer
v=2ti+3j\mathbf{v}=2 t\mathbf{i}+3\mathbf{j}
Question 2
2 markseasy
A particle moves in a plane so that its position vector at time tt seconds is r=2t2i+tj\mathbf{r} = 2 t^{2}\mathbf{i}+t\mathbf{j} metres. Find its velocity vector v\mathbf{v} as a function of tt.

Worked solution

  1. Write down the position vector

    r=2t2i+tj\mathbf{r}=2 t^{2}\mathbf{i}+t\mathbf{j}

    The motion is described by this position vector; velocity is its rate of change.

  2. Differentiate each component with respect to time

    drdt=4ti+1j\frac{d\mathbf{r}}{dt}=4 t\mathbf{i}+1\mathbf{j}

    Differentiating each component gives the velocity vector.

  3. State the velocity vector

    v=4ti+1j\mathbf{v}=4 t\mathbf{i}+1\mathbf{j}

    This is the velocity as a function of time.

Answer
v=4ti+1j\mathbf{v}=4 t\mathbf{i}+1\mathbf{j}
Question 3
2 markseasy
A particle moves in a plane so that its position vector at time tt seconds is r=t2−2ti+4j\mathbf{r} = t^{2} - 2 t\mathbf{i}+4\mathbf{j} metres. Find its velocity vector v\mathbf{v} as a function of tt.

Worked solution

  1. Write down the position vector

    r=t2−2ti+4j\mathbf{r}=t^{2} - 2 t\mathbf{i}+4\mathbf{j}

    The motion is described by this position vector; velocity is its rate of change.

  2. Differentiate each component with respect to time

    drdt=2t−2i\frac{d\mathbf{r}}{dt}=2 t - 2\mathbf{i}

    Differentiating each component gives the velocity vector.

  3. State the velocity vector

    v=2t−2i\mathbf{v}=2 t - 2\mathbf{i}

    This is the velocity as a function of time.

Answer
v=2t−2i\mathbf{v}=2 t - 2\mathbf{i}
Question 4
2 markseasy
A particle moves in a plane so that its position vector at time tt seconds is r=5ti+2t2j\mathbf{r} = 5 t\mathbf{i}+2 t^{2}\mathbf{j} metres. Find its velocity vector v\mathbf{v} as a function of tt.

Worked solution

  1. Write down the position vector

    r=5ti+2t2j\mathbf{r}=5 t\mathbf{i}+2 t^{2}\mathbf{j}

    The motion is described by this position vector; velocity is its rate of change.

  2. Differentiate each component with respect to time

    drdt=5i+4tj\frac{d\mathbf{r}}{dt}=5\mathbf{i}+4 t\mathbf{j}

    Differentiating each component gives the velocity vector.

  3. State the velocity vector

    v=5i+4tj\mathbf{v}=5\mathbf{i}+4 t\mathbf{j}

    This is the velocity as a function of time.

Answer
v=5i+4tj\mathbf{v}=5\mathbf{i}+4 t\mathbf{j}
Question 5
2 markseasy
A particle moves in a plane with velocity v=4ti+6j\mathbf{v} = 4 t\mathbf{i}+6\mathbf{j} (m s−1^{-1}) at time tt seconds. Find its acceleration vector a\mathbf{a} as a function of tt.

Worked solution

  1. Write down the velocity vector

    v=4ti+6j\mathbf{v}=4 t\mathbf{i}+6\mathbf{j}

    Acceleration is the derivative of velocity with respect to time.

  2. Differentiate each component

    dvdt=4i\frac{d\mathbf{v}}{dt}=4\mathbf{i}

    Differentiating each component gives the acceleration vector.

  3. State the acceleration vector

    a=4i\mathbf{a}=4\mathbf{i}

    This is the acceleration as a function of time.

Answer
a=4i\mathbf{a}=4\mathbf{i}

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