12-MATHS-C
PRSHS
Document 1103(a):Worksheet 1- Vector Calculus
Questions requiring differentiation:
Question 1:
The position vectors r(t) are given below in terms of t. Find the velocity and acceleration vectors:
(a) r(t) = 2t i +10t2 j
(b) r(t) = 4t3 i –(5t-t2) j
Question 2:
The position vector of a particle P is given by (2t  t 2 )i  (3  t ) j ,
where t is the time elapsed in seconds.
(i)
From what point does the particle P begin to move?
(ii)
What is the initial velocity and direction of movement?
(iii)
After 2 seconds where is P and what is the velocity and direction of
movement?
Question 3:
A particle moves so that its displacement after “t” seconds is ( 3t 2  2t  10)i  ( t 3  2t  6) j
Describe the position and velocity of the particle after one second has elapsed.
Question 4:
An object is projected so that its position at any time t seconds is given by:
r = 2t i + (30t-5t2) j
(measured in metres)
Find:
(a) the initial point of projection of the object.
(b) the displacement after 2 seconds and the distance from the starting point at this time.
(c) the speed at the initial projection and the angle of projection
(d) the speed after 2 seconds
(e) the acceleration vector and the magnitude of acceleration.
(f) the time taken to hit the ground
(g) the horizontal distance travelled
(h) the maximum height reached
(i) the cartesian equation for the path of the object
Question 5:
An object is projected so that its position at any time t seconds is given by:
r = 4t i + (6-5t2) j
(measured in metres)
(a) From what position is the object projected
(b) Calculate the speed of projection
(c) At what angle to the horizontal is it projected?
(d) Show that the acceleration is constant.
Question 6:
A particle moves with a displacement vector at any time t seconds given by: r1  3t i  (2t  3) j . A
2
second particle has displacement vector given by r2  (t  4)i  (t  1) j at any time t seconds.
(a) show that the two particles collide .
(b) find the time and place of the collision
Questions involving Integration
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Question 7:
An object moves in a straight line so that its acceleration at t seconds is given by:
a = (3-4t) i
Initially the object has a speed of 2m/s in the i direction and is 5 metres from a fixed point O.
(a) Calculate its velocity and position after t seconds.
(b) Where and when is the body at rest.
Question 8:
The acceleration of a particle at any time t (seconds) is given by:
a (t) = 2t i - 3 j
The initial velocity is given by:
v = -4 i +6 j
(a) Find an expression for the velocity of the particle at any time t.
(b) When is the velocity zero?
(c ) What is the magnitude of its acceleration at this time?
Question 9:
The velocities of two particles P and Q are given by
v P  2 i  2t j
and
vQ  3 i  5 j
At t = 0, the positions of the particles are
rP  5 i  6 j and
rQ   i
(a) Find the position of each particle at any time “t’
(b) Where and when do the particles collide?
(c ) Calculate the Cartesian path of each particle
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