Chapter 4 - MOTION IN A PLANE

A river is flowing due east with a speed 3m/s. A
swimmer can swim in still water at a speed of 4 m/s
(Fig. 4.8).
(a) If swimmer starts swimming due north, what will
be his resultant velocity (magnitude and direction)?
(b) If he wants to start from point A on south bank and
reach opposite point B on north bank,
(a) which direction should he swim?
(b) what will be his resultant speed?
(c) From two different cases as mentioned in (a) and (b)
above, in which case will he reach opposite bank in
shorter time?

A girl riding a bicycle with a speed of 5 m/s towards north direction,
observes rain falling vertically down. If she increases her speed to
10 m/s, rain appears to meet her at 45° to the vertical. What is the
speed of the rain? In what direction does rain fall as observed by a
ground based observer?

A particle is projected in air at an angle
b to a surface which itself is inclined at
an angle a to the horizontal (Fig. 4.6).
(a) Find an expression of range on the
plane surface (distance on the plane
from the point of projection at which
particle will hit the surface).
(b) Time of flight.
(c) b at which range will be maximum.

4.29 A hill is 500 m high. Supplies are to be sent across the hill using a
canon that can hurl packets at a speed of 125 m/s over the hill.
The canon is located at a distance of 800m from the foot of hill and
can be moved on the ground at a speed of 2 m/s; so that its distance
from the hill can be adjusted. What is the shortest time in which a
packet can reach on the ground across the hill ? Take g =10 m/s2

A fighter plane is flying horizontally at an altitude of 1.5 km with
speed 720 km/h. At what angle of sight (w.r.t. horizontal) when
the target is seen, should the pilot drop the bomb in order to
attack the target?

In dealing with motion of projectile in air, we ignore effect of air
resistance on motion. This gives trajectory as a parabola as you
have studied. What would the trajectory look like if air resistance
is included? Sketch such a trajectory and explain why you have
drawn it that way.

A boy throws a ball in air at 60° to the horizontal along a road with
a speed of 10 m/s (36km/h). Another boy sitting in a passing by
car observes the ball. Sketch the motion of the ball as observed by
the boy in the car, if car has a speed of (18km/h). Give explanation
to support your diagram.

A boy travelling in an open car moving on a levelled road with
constant speed tosses a ball vertically up in the air and catches it
back. Sketch the motion of the ball as observed by a boy standing
on the footpath. Give explanation to support your diagram.

A, B and C are three non-collinear, non co-planar vectors.What
can you say about direction of A × (B × C)?

A football is kicked into the air vertically upwards. What is its
(a) acceleration, and (b) velocity at the highest point?

A ball is thrown from a roof top at an angle of 45° above the
horizontal. It hits the ground a few seconds later. At what point
during its motion, does the ball have
(a) greatest speed.
(b) smallest speed.
(c) greatest acceleration?
Explain

At what point of projectile motion (i) potential energy maximum (ii) Kinetic energy
maximum (iii) total mechanical energy is maximum

A body of mass m is thrown with velocity 'u ' at angle of 30^o to the horizontal and

another body B of the same mass is thrown with velocity u at an angle of 60^o to the

horizontal. Find the ratio of the horizontal range and maximum height of A and B?

What is a uniform circular motion? Explain the terms time period, frequency and
angular velocity. Establish relation between them.

The blades of an aeroplane propeller are rotating at the rate of 600 revolutions per
minute. Calculate its angular velocity.

What are the two angles of projection of a projectile projected with velocity 30m/s,

so that the horizontal range is 45m. Take, g = 10m/s².

A stone tied at the end of string is whirled in a circle. If the string breaks, the stone
flies away tangentially. Why?

A body is moving on a curved path with a constant speed. What is the nature of its
acceleration?

What is the angular velocity of the hour hand of a clock?

What will be the effect on maximum height of a projectile when its angle of

projection is changed from 30^o to 60^o, keeping the same initial velocity of

projection?

What will be the effect on horizontal range of a projectile when its initial velocity is
doubled, keeping the angle of projection same?

Why does the direction of motion of a projectile become horizontal at the highest
point of its trajectory?

Which is greater the angular velocity of the hour hand of a watch or angular
velocity of earth around its own axis?

Two forces 5 and 10 kg wt are acting with an inclination of 120^o between them.

What is the angle which the resultant makes with 10kg wt?

Two forces 5 and 10 kg wt are acting with an inclination of 120^o between them.

What is the angle which the resultant makes with 10kg wt?

What is the direction of centripetal force when particle is following a circular path?

Give an example of a body moving with uniform speed but having a variable
velocity and an acceleration which remains constant in magnitude but changes in
direction

(a) Define time of flight and horizontal range?
(b) From a certain height above the ground a stone A is dropped gently.
Simultaneously another stone B is fired horizontally. Which of the two stones will
arrive on the ground earlier?

Derive an equation for the path of a projectile fired parallel to horizontal.

Derive expressions for velocity and acceleration for uniform circular motion.
OR Derive expression for linear acceleration in uniform circular motion.

At what range will a radar set show a fighter plane flying at 3 km above its centre
and at distance of 4 km from it?

Prove that for elevations which exceed or fall short of 45^o by equal amounts the

ranges are equal?

What is the angle of projection at which horizontal range and maximum height are
equal?

What is the angle between two forces of 2N and 3N having resultant as 4N?

What is the value of angular speed for 1 revolution?

A projectile is fired at an angle of 30^o with the horizontal with velocity 10m/s. At

what angle with the vertical should it be fired to get maximum range?

What is “Trajectory of a projectile?

(a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by
θ(t) = tan^-1((v_0y- gt) / v_0x)
(b) Shows that the projection angle θ⁰ for a projectile launched from the origin is given by
θ₀ = tan^-1(4h_m / R)
where the symbols have their usual meaning.

A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s^-1 to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit ? (Take g = 10 m s^-2 ).

 A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to the fixed, and neglect air resistance.

Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere ? Explain.

 A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector ? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector ?

A vector has magnitude and direction. Does it have a location in space ? Can it vary with time ? Will two equal vectors a and b at different locations in space necessarily have identical physical effects ? Give examples in support of your answer.

An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the speed of the aircraft ?

Read each statement below carefully and state, with reasons and examples, if it is true or false :
A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes.

 For any arbitrary motion in space, which of the following relations are true :
(a) v_average = (1/2) (v (t₁) + v (t₂))
(b) v_average = [ r(t₂) - r(t₁) ] / (t₂ – t₁)
(c) v (t) = v (0) + a t
(d) r (t) = r (0) + v (0) t + (1/2) a t²
(e) a_average = [ v (t₂) - v (t₁ ) ] / ( t₂ – t₁)
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

 î and ĵ are unit vectors along x and y axis respectively. What is the magnitude and direction of the vectors î + ĵ, and î − ĵ ? What are the components of a vector A= 2î + 3ĵ along the directions of î + ĵ and î − ĵ? [You may use graphical method]

A particle starts from the origin at t = 0 s with a velocity of 10.0 ĵ m/s and moves in the x-y plane with a constant acceleration of (8.0 î+ 2.0 ĵ) 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

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?

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 man can swim with a speed of 4.0 km/h in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?

Rain is falling vertically with a speed of 30 m s^-1. A woman rides a bicycle with a speed of 10 m s^-1 in the north to south direction. What is the direction in which she should hold her umbrella ?

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?

On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

A cyclist starts from the centre Oof a circular park of radius 1 km, reaches the edge Pof the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist?

Three girls skating on a circular ice ground of radius 200 m start from a point Pon the edge of the ground and reach a point Qdiametrically opposite to Pfollowing different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of the path skated?

Given vectors a + b + c + d = 0, which of the following statements are correct:
(a) Vectors a, b, c, and d must each be a null vector,
(b) The magnitude of vectors (a + c) equals the magnitude of vectors(b+ d),
(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,
(d) Vectors b + c must lie in the plane of a and d if  a and d are not collinear, and in the line of a and d, if they are collinear?

Establish the following vector inequalities geometrically or otherwise:
(a) |a + b| ≤ |a| + |b|
(b) |a + b| ≥ ||a| − |b||
(c) |a − b| ≤ |a| + |b|
(d) |a − b| ≥ ||a| − |b||
When does the equality sign above apply?

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:
(a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions, (c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding acomponent of a vector to the same vector.

 Pick out the only vector quantity in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

 State, for each of the following physical quantities, if it is a scalar or a vector:
volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.