Mechanics 1 Modelling with Friction Question 11

A chute at a water sports centre has been designed so that swimmers first slide down a steep part which is 10 m long and at an angle of 40° to the horizontal. They then come to a 20 m section with a gentler slope, 11° to the horizontal, where they travel at constant speed.

(i) Find the coefficient of friction between a swimmer and the chute.

(ii) Find the acceleration of a swimmer on the steep part.

(iii) Find the speed at the end of the chute of a swimmer who starts at rest.

(You may assume that no speed is lost at the point where the slope changes.)

An alternative design of chute has the same starting and finishing points but has a constant gradient.

(iv) With what speed do swimmers arrive at the end of this chute?

Solution:

9709/MayJune/1/2008/Q11

In the diagram, the points A and C lie on the x- and y-axes respectively and the equation of AC is
2y + x = 16. The point B has coordinates (2, 2). The perpendicular from B to AC meets AC at the
point X.
(i) Find the coordinates of X. [4]
The point D is such that the quadrilateral ABCD has AC as a line of symmetry.
(ii) Find the coordinates of D. [2]
(iii) Find, correct to 1 decimal place, the perimeter of ABCD. [3]

Solution:

(ii)

D = (6,10)

(iii)

Reference: PYQ - May/June 2008 Paper 1 Q11

9709/MayJune/12/2010/Q11

The function f : x → 4 − 3 sin x is defined for the domain 0 ≤ x ≤ 2pi.

(i) Solve the equation f(x) = 2. [3]

(ii) Sketch the graph of y = f(x). [2]

(iii) Find the set of values of k for which the equation f(x) = k has no solution. [2]

The function g : x → 4 − 3 sin x is defined for the domain 1/2 ≤ x ≤ A

(iv) State the largest value of A for which g has an inverse. [1]

(v) For this value of A, find the value of g⁻¹(3). [2]

Solution:

i)

ii)

f(x) = 3 sin x

f(x) = -3 sin x

f(x) = 4 - 3 sin x

iii)

iv)

v)

Reference: PYQ - May/June 2010 Paper 12 Q11