Motion in a Plane Test 5

As the boat moves at 10 kph across the river it is being swept downstream by the river current at a 90 degree angle. resulting i a motion of 14 kph. the downstream velocity of the river can be calculated by the Pythagoras theorem:

10^2+x^=14^2

100+x^2=196

x^2=96

x=9.798 approximately

Hour hand i.e- 60x60=3600sec
Time period=2π/angular speed.
T=2π/ ω
ω=2π/60x60=2π/3600
to complete one rotation, time taken=12 hours.
For 12 hours=12x60x60 sec
=43200 sec
Angular speed of hour hand=2π/T
(i.e., a complete round) =2π/43200
= π/21600 rad/s

When the man rides, he should put an umbrella in the opposite direction to which the drops are falling with respect to him.
Hence wrt man, the speed of rain drop is
at an angle of tan^-1 ⅓
Hence he would put umbrella to the direction of tan^-1 ⅓

Radius r = 0.5 m
Speed, v = 20 m/s
We know,
v = r ω
where, 
ω = angular velocity
20 = 0.5 x ω
ω = 20/0.5 = 40 rad/sec

Correct Answer :- d

Explanation : Let the initial velocity of the body be u m/s; and the constant acceleration be a m/s². Then displacement (distance covered) of the body after t second is given by

s = u t + ½ a t²       —(1)

The body covers 50 m in 5s and 80 m in 10s.

50 = u×5 + ½ a 5²; => 50 = 5u + 25/2a

10u + 25a = 100     —(2)

80 = u×10 + ½ a 10²; => 80 = 10u + 50a; this gives,

u + 5a = 8  —(3)

Solving (2) and (3) for u and a we get

u = 12 m/s and a = -4/5 m/s²  — (4)

To obtain the distance covered in the next 15s we need to find the velocity at the beginning of this interval using,

v = u + a t => v = 12 m/s + (-4/5) m/s² × 15s = 0m/s

s = u t + ½ a t²

Substituting various values we get

s = 12×15 +½ ×(-4/5)×15² = 180 - 90

= 90 m

As the flywheel makes 210 revolution in a minute, and during each revolution the angular displacement of the flywheel  is 2π thus total angular distance covered is
 210 × 22/7 × 2 = 1320 rad
Therefore angular speed=1320 rad/60 s = 22 rad/s

Velocity of boat wrt river = velocity of boat wrt ground - velocity of river wrt ground
I.e. v = (6i + 8j) - (-6i - 4j)
= 12i + 12j

The second hand goes through 2π radians in 1 min, or 2π radian/60 seconds, so ω = π/30 rad.s^-1 = 0.03 rad.s^-1

Assuming the direction of velocity of train to be positive, wrt train the speed of monkey will be
-100 + 5 = -95 kmph
I.. 95 kmph in direction opposite to the train.

Radius r = 0.5 m

Speed, v = 20 m/s

We know,

v = r ω

where, 

ω = angular velocity

20 = 0.5 x ω

ω = 20/0.5 = 40 rad/sec