A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s². Find (a) the angle between the acceleration and i, (b) the magnitude of F. At time t seconds the velocity of P is v m s⁻¹. Given that when t = 0, v = 9i – 10j, (c) find the velocity of P when t = 5.

A-Level Edexcel Maths Mechanics Working with Vectors: A particle P of mass 0.4 kg move

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1

Question 3

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s². Find (a) the angle between the acce... show full transcript

Worked Solution & Example Answer:A particle P of mass 0.4 kg moves under the action of a single constant force F newtons - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1

Step 1

the angle between the acceleration and i

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Answer

To find the angle θ between the acceleration vector and the i direction, we first recognize that:

$a = 6i + 8j$

Using the formula for the tangent of the angle:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

We can find:

$\tan(\theta) = \frac{8}{6}$

Thus,

$\theta = \tan^{-1}\left(\frac{8}{6}\right) \approx 53°$

Step 2

the magnitude of F

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Answer

The force F can be determined using Newton's second law, which states:

$F = ma$

Where m is the mass (0.4 kg) and a is the acceleration vector (6i + 8j). Hence,

$F = 0.4(6i + 8j) = (2.4i + 3.2j)$

To find the magnitude of F:

$|F| = \sqrt{(2.4)^2 + (3.2)^2} = \sqrt{5.76 + 10.24} = \sqrt{16} = 4$

Step 3

find the velocity of P when t = 5

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Answer

Starting from the initial velocity at t = 0:

$v(0) = 9i - 10j$

The acceleration we have is:

$a = 6i + 8j$

The velocity after time t is given by:

$v(t) = v(0) + at$

Substituting for t = 5 seconds:

$v(5) = (9i - 10j) + (6i + 8j) * 5$

This gives:

$v(5) = 9i - 10j + (30i + 40j)$

Thus,

$v(5) = (9 + 30)i + (-10 + 40)j = 39i + 30j \text{ m/s}$

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1

Worked Solution & Example Answer:A particle P of mass 0.4 kg moves under the action of a single constant force F newtons - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1

the angle between the acceleration and i

the magnitude of F

find the velocity of P when t = 5

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