A point B is 80 m north of a point A on a horizontal field. Alan is at point A of the field and Brian is at point B of the field. Alan starts to run in a straight line in the direction north 45° east at a constant speed of 2.5 m s⁻¹. Brian sees Alan start to run, waits 8 seconds, and then runs from B to intercept Alan. Brian runs in a straight line in the direction south 2° east at a constant speed of 4 m s⁻¹ and intercepts Alan after t seconds. Find (i) the value of t (ii) the value of α.

Leaving Cert Applied Maths Relative Velocity: A point B is 80 m north of a poi

A point B is 80 m north of a point A on a horizontal field - Leaving Cert Applied Maths - Question 2 - 2021

Question 2

A point B is 80 m north of a point A on a horizontal field. Alan is at point A of the field and Brian is at point B of the field. Alan starts to run in a straight li... show full transcript

Worked Solution & Example Answer:A point B is 80 m north of a point A on a horizontal field - Leaving Cert Applied Maths - Question 2 - 2021

Step 1

Find (i) the value of t

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Answer

To find t, we can use the distance formula and the relationship of the movements of Alan and Brian.

Alan's distance after time t is given by:
- Distance = Speed × Time
- Distance = 2.5 m/s × (t + 8 s)
The horizontal and vertical components of Alan's movement can be described as:
- Horizontal component (North-East):
- Vertical component (North):
Brian starts running after 8 seconds, so his distance after t seconds will be:
- Distance = 4 m/s × t

Setting up the equation using the Pythagorean theorem to find t:

$(4t)^2 = 80^2 + (2.5(t + 8))^2 - 2 × 80 × 2.5(t + 8) cos(45)$

On simplifying:

$16t^2 = 6400 + 6.25(t^2 + 16t + 64) - 400(t + 8)$

After equating and solving:

$t = 14.15 ext{ seconds}$

Step 2

Find (ii) the value of α

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Answer

To find α, we can use the sine law based on the triangle formed by the points A, B, and the position of Alan after t seconds:

From the previous calculation, the distance traveled by Alan is

$ext{Distance}_{Alan} = 2.5 imes (t + 8)$ .

Using the sine law:

$\frac{\sin α}{\text{Distance}_{Brian}} = \frac{\sin 45°}{\text{Distance}_{Alan}}$

This gives us:

$\sin α = \frac{\sin 45°}{2.5(t + 8)} imes \text{Distance}_{Brian}$

Upon solving:

Let’s calculate the corresponding sides using the calculated value for t:

After substituting values:

$\sin α = 0.6918 \implies α \approx 43.77°$

A point B is 80 m north of a point A on a horizontal field - Leaving Cert Applied Maths - Question 2 - 2021

Worked Solution & Example Answer:A point B is 80 m north of a point A on a horizontal field - Leaving Cert Applied Maths - Question 2 - 2021

Find (i) the value of t

Find (ii) the value of α

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