For any arbitrary motion in space, which of the following relations are true :
(a) vaverage
= (1/2) (v (t
1
) + v (t
2
))
(b) v
average
= [r(t
2
) – r(t
1
) ] /(t
2
– t
1
)
(c) v (t) = v (0) + a t
(d) r (t) = r (0) + v (0) t + (1/2) a t
2
(e) a average
=[ v (t
2
) – v (t
1
)] /( t
2
– t
1
)
(The ‘average’ stands for average of the quantity over the time interval t
1
to t
2
)
ɵ
i
and ɵ
j are unit vectors along x- and y- axis respectively. What is the magnitude
and direction of the vectors ɵ ɵ
i j + , and ɵ ɵ
i j − ? What are the components of a vector
A= 2 ɵ ɵ
i j + 3 along the directions of ɵ ɵ
i j + and ɵ ɵ
i j − ? [You may use graphical method]
A particle starts from the origin at t = 0 s with a velocity of 10.0 jɵ m/s and moves in
the x-y plane with a constant acceleration of (8.0 2.0 )
ɵ ɵ
i j + m s-2. (a) At what time is
the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at
that time? (b) What is the speed of the particle at the time ?
The position of a particle is given by
ˆ ˆ ˆ 2 r i j k = − + 3.0 2.0 4.0 m t t
where t is in seconds and the coefficients have the proper units for r to be in metres.
(a) Find the v and a of the particle? (b) What is the magnitude and direction of
velocity of the particle at t = 2.0 s ?
Read each statement below carefully and state, with reasons, if it is true or false :
(a) The net acceleration of a particle in circular motion is always along the radius of
the circle towards the centre
(b) The velocity vector of a particle at a point is always along the tangent to the path
of the particle at that point
(c) The acceleration vector of a particle in uniform circular motion averaged over one
cycle is a null vector
An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900
km/h. Compare its centripetal acceleration with the acceleration due to gravity.
A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a
constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and
direction of acceleration of the stone ?
A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much
high above the ground can the cricketer throw the same ball ?
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that
a ball thrown with a speed of 40 m s-1 can go without hitting the ceiling of the hall ?
A passenger arriving in a new town wishes to go from the station to a hotel located
10 km away on a straight road from the station. A dishonest cabman takes him along
a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average
speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ?
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