(a) P(-1, 4), Q(2, 3), R(x, y) and S(-2, 3) are the verticles of a parallelogram. Find the value of x and y. (b) A particle starts from rest and moves in a ...

Answer Details

(a) To find the value of x and y, we can use the fact that opposite sides of a parallelogram are parallel and have equal length.

From the given information, we know that PQ and SR are parallel, and PQ = SR since they are opposite sides of the parallelogram. Therefore, we can use the distance formula to find the length of PQ and SR:

PQ = sqrt((2 - (-1))^2 + (3 - 4)^2) = sqrt(3^2 + (-1)^2) = sqrt(10)

SR = sqrt((-2 - x)^2 + (3 - y)^2)

Since PQ = SR, we have:

sqrt(10) = sqrt((-2 - x)^2 + (3 - y)^2)

Squaring both sides, we get:

10 = (2 + x)^2 + (3 - y)^2

Expanding the squares, we get:

10 = 4 + 4x + x^2 + 9 - 6y + y^2

Simplifying, we get:

x^2 + y^2 + 4x - 6y - 3 = 0

We can rewrite this equation as:

(x + 2)^2 + (y - 3)^2 = 13

This is the equation of a circle with center (-2, 3) and radius sqrt(13). Therefore, any point (x, y) that lies on this circle will form a parallelogram with P(-1, 4), Q(2, 3), and S(-2, 3).

(b)

(i) We know that the initial velocity of the particle is zero, and that its final velocity is 20ms^-1 after travelling a distance of 8 metres. We can use the formula:

v^2 = u^2 + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance travelled.

Plugging in the given values, we get:

(20)^2 = 0^2 + 2a(8)

Simplifying, we get:

a = 100/8 = 12.5ms^-2

Therefore, the acceleration of the particle is 12.5ms^-2.

(ii) We can use the formula:

s = ut + 0.5at^2

where s is the distance travelled, u is the initial velocity, a is the acceleration, and t is the time taken. Since the initial velocity is zero, the formula simplifies to:

s = 0.5at^2

Plugging in the given values, we get:

40 = 0.5(12.5)t^2

Simplifying, we get:

t^2 = 6.4

Taking the square root of both sides, we get:

t = 2.53 seconds (rounded to two decimal places)

Therefore, the time taken