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QUESTION 19 (7 marks) An object is swinging at the end of a 0.5 m length of string in a vertical circular path with a constant angular speed completing each revolution in 0.24 seconds. The object is projected from a height of 0.3 m above the ground in a vertical plane and just passes over a narrow pole as shown in the diagram. The pole is 2.05 m high and its base is 14 m horizontally from where the object was projected. _______ iy . ~. Not drawn to scale ~ 14 m 0.6 m A flat-topped vehicle of length 0.6 m and height 0.4 m is initially at rest against the pole as shown in the diagram. At the instant that the object is projected the vehicle moves in a horizontal direction away from the pole in the same vertical plane with an acceleration of magnitude of @ m s 2. The object strikes the middle of the top of the vehicle. Assuming that air resistance is negligible use vector calculus to model the motion of the projectile in order to determine the value of a.
QUESTION 3 An object has a velocity v(7) = e i+ G]fi where ¢ represents time (7 > 0). The displacement r(7) of the object could be (A) —2ei+In(r)k -2t % (B) —2¢ i —LZIG
QUESTION 14 (6 marks) An object is projected vertically upwards from ground level. After the object has been in motion for t seconds its position vector through the air in metres is modelled by A . r(t)=5t(8-1)j a) Determine the velocity of the object through the air v (t) in metres per second. [2 marks] b) Determine the number of seconds until the object reaches its maximum height. [2 marks] c) Determine the maximum height that the object reaches in metres. [2 marks]
QUESTION 19 (7 marks) The velocity vectors of two objects A and B (in m s_l) at time ¢ (in s) are given respectively by v = 6sin(3t)i +6cos(3t) j v = cos(¢)i —sin(¢)j Objects A and B are initially at (-2 0 2) and (0 1 — 1) respectively. Determine the position of Object A when it is 4 metres away from Object B for the first time.
QUESTION 3 A particle travels in a straight line over time ¢ with a constant acceleration a(t). Which function could represent the particle’s displacement x(t)? (A) x(t)=7 (B) x(t)=t* 1 (C) X(t) = ; D) x(¢)=+r
QUESTION 11 (6 marks) The position vector of a particle r|(cm) over time ¢(s) is given by r ()= (2t +1)i +(¢+3) j— (2t -3)k a) Determine the velocity vector of the particle. [1 mark] b) Determine the time when the position vector of the particle is perpendicular to its velocity vector. [2 marks] The position vector of a second particle rz(cm) over time t(s) is given by ry(t)=(16—4t)i —(3t—13)j+ 2k c) Determine whether the two particles collide. [3 marks]
QUESTION 3 Which of the following equations describes circular motion? (A) r(t) =sin(t) T+ cos(2t)j (B) 1(t) = 2cos(t)i+ sin(t)jf (C) r(t)=2cos(t)i+sin(2t)j (D) r(t) =2cos(2t) i+ 2sin(2t) ]
QUESTION 12 (10 marks) The diagram below shows a ramp inclined at an angle of 30° to the horizontal. A trolley of mass 20 kg is being pushed up the ramp by a worker with a net force parallel to the plane. Not drawn to scale a) Sketch and appropriately label all forces acting on the trolley in this situation on the diagram below. Assume there are negligible frictional forces and no air resistance. [2 marks] . \. Not drawn to scale b) Determine the minimum force required by the worker to prevent the trolley from moving down the ramp. [2 marks] c) Ifthe worker exerts a force of 178 N on the trolley determine the acceleration of the trolley up the ramp. [3 marks] A worker pulls the trolley up the same ramp with a net force parallel to the plane as shown in the diagram below. t The velocity v (m s~1) of the trolley at time t (s) is given by v = S m sTH(0<t<55). After 2 seconds the trolley is positioned 5 m from the bottom of the ramp. Not drawn to scale d) Determine the position of the trolley from the bottom of the ramp after 4 seconds. [3 marks]
QUESTION 16 (8 marks) The general form of an ellipse in Cartesian form is (x —h) +(y—k) a? b2 =1 The path of an object with a position function of r(t) = cos?(t) i + sin(2t) j is stated to be elliptical. Evaluate the reasonableness of this statement.
QUESTION 1 The position of a particle is given by r= (t+ 2)i + tj for t>0. Determine the corresponding Cartesian equation. (A) yax-4 (B) y=x°+4 (C) y=x —4x+4 (D) y=x*+4x+4
QUESTION 19 (6 marks) Object A is released from the origin with constant velocity v such that its position after t seconds is given by r =2V3tit+3tj+2tk t>0. At a later time object B is released from point P (3V3 6 0) and travels towards point 0(5V3 8 4) with constant velocity vp such that Iv al =/2 lv Al: Given that objects A and B collide determine the time between the release of the two objects. Assume all positions are given in metres and all velocities are given in metres per second.
QUESTION 4 The position of a particle can be modelled using r = cos(t)i—2sin(t) j 20. Which curve best represents the path of the particle? (A) y (B) (C) y (D)
QUESTION 17 (6 marks) An object is projected upwards from ground level with an initial velocity of 15 m sat an angle of 54° to the horizontal. The object just passes over a drone hovering in the air. An observer is positioned directly below the drone and at a horizontal distance of 20 m from where the object is projected. The observer commented that: ¢ it took the object around 2 to 2.5 seconds after its projection to reach the drone ¢ the object was still moving in an upwards direction as it passed the drone. Assuming that air resistance is negligible use a vector calculus approach to evaluate the reasonableness of the observer’s comments.
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