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Two forces (4i - 2j) N and (2i + qj) N act on a particle P of mass 1.5 kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2014 - Paper 1
Question 2
Two forces (4i - 2j) N and (2i + qj) N act on a particle P of mass 1.5 kg. The resultant of these two forces is parallel to the vector (2i + j). (a) Find the value ... show full transcript
Worked Solution & Example Answer:Two forces (4i - 2j) N and (2i + qj) N act on a particle P of mass 1.5 kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2014 - Paper 1
Step 1
Find the value of q.
Answer
To determine the value of q, we start by calculating the resultant force from both forces:
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The first force,
F₁ = (4i - 2j) N
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The second force,
F₂ = (2i + qj) N
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The resultant force, F_R, is given by:
F_R = F₁ + F₂ = (4i - 2j) + (2i + qj) = (6i + (q - 2)j) N.
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This resultant force needs to be parallel to the vector (2i + j). Thus, we can express this parallelism through a scalar multiple:
(6i + (q - 2)j) = k(2i + j), where k is a scalar.
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By comparing the components:
6 = 2k
and
(q - 2) = k.
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From the first equation, solving for k yields:
k = 3.
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Substituting k back into the second equation to find q:
(q - 2) = 3
implies that q = 5.
Thus, the value of q is 5.
Step 2
Find the speed of P at time t = 2 seconds.
Answer
At t = 0, the particle P has an initial velocity:
v₀ = (-2i + 4j) m s⁻¹.
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The resultant acceleration (a) is given by the net force divided by the mass:
a = F_R / m, where m = 1.5 kg.
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We first need to determine the resultant force (calculated previously):
F_R = (6i + 3j) N, (since q = 5).
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Therefore, the acceleration is:
a = (6i + 3j) / 1.5 = (4i + 2j) m s⁻².
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The velocity after time t can be calculated using the equation:
v = v₀ + at, where t = 2 seconds:
v = (-2i + 4j) + ((4i + 2j) * 2).
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Simplifying this gives:
v = (-2i + 4j) + (8i + 4j) = (6i + 8j) m s⁻¹.
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To find the speed, we calculate the magnitude of the velocity:
speed = ||v|| =
= \sqrt{(6^2 + 8^2)} = \sqrt{36 + 64} = \sqrt{100} = 10 m s⁻¹.
Thus, the speed of P at time t = 2 seconds is 10 m s⁻¹.