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Q.4.21 A particle starts from the origin at t = 0 s with a velocity of 10.0 ? m/s and moves in the x-y plane with a constant acceleration of (8.0 ? + 2.0 ?) m s-2.
(a) At what time is the x- coordinates 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?
Q.4.21 A particle starts from the origin at t = 0 s with a velocity of 10.0 ? m/s and moves in the x-y plane with a constant acceleration of (8.0 ? + 2.0 ?) m s-2.
(a) At what time is the x- coordinates 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?
Ans.4.21: (a) Velocity , = 10.0 ? m/s Acceleration, = (8.0 ? + 2.0 ?) m s-2 We know = = 8.0 ? + 2.0 ? = (8.0 ? + 2.0 ?)dt Integrating both sides we get (t) = 8.0t ? + 2.0t ? + , Where, velocity vector of the particle at t =0 velocity vector of the particle at time t But = = dt = (8.0t ? + 2.0t ? + )dt Integrating both sides with the condition at t = 0, r =0 and at t =t, r = r t + ½ 8.0 t2 ? + ½ 2.0 t2 ? = t + 4.0 t2 ? + t2 ? Substituting the value of , we get ( 10.0 ?)t + 4.0 t2 ? + t2 ? . This equation can be expressed as x ? + y ? = 4.0 t2 ? + ( 10.0t + t2) ? Since the motion of the particle is confined to the x-y plane, on equating the coefficients of ? and ?, we get x = 4.0 t2 and y = 10.0t + t2 t = (a) When x = 16m, t = 2 s, y = 24m (b) Velocity of the particle (t) = 8.0t ? + 2.0t ? + At t = 2 s, (t) = 8.0 2 ? + 2.0 ? + 10 ? = 16 ? + 14 ? The magnitude of (t) is given by = ( 162 + 142)1/2 = 21.26 m/s |
4.25 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?
4.25 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?
4.25

Height of the aircraft from the ground = 3400m
Let A and B be the positions of the aircraft, making an angle of AOB = 30 . The perpendicular OC and OC is the height of the aircraft. Angles AOC = angle BOC = 15
In AOC, AC = OC
= 3400 = 911.03m
AB = AC + CB = 2AC = 1
24 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
24 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
4.24
(a) Despite being a scalar quantity, energy is not conserved in elastic collisions. False
(b) Despite being a scalar quantity, the temperature can take negative values. False
(c) The total path length is a scalar quantity. Yet it has the dimension of length. False
(d) A scalar quantity such as
4.23 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)
4.23 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)
4.23
(a) For any arbitrary motion of a particle average velocity cannot be expressed by this equation. False
(b) The arbitrary motion of the particle can be represented by this equation, True
(c) For arbitrary motion of the particle, the acceleration may also be non uniform. False
What is the significance of de Broglie’s hypothesis in modern physics?
What is the significance of de Broglie’s hypothesis in modern physics?
Significance of de Broglie's Hypothesis in Modern Physics (NCERT-based answer):
De Broglie's hypothesis is a revolutionary idea which changed the understanding of matter and wave as we know it. This hypothesis introduced the concept that particles of matter, like electrons, and Proton, can behave lik
2.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 kg m2 s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α –1 β –2 γ 2 in terms of the new units.
2.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 kg m2 s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α –1 β –2 γ 2 in terms of the new units.
2.3
1 Calorie = 4.2 J = 4.2 kgm2s-2
Standard formula for conversion
Given unit / New unit = (M1/M2)x (L1/L2)y (T1/T2)z
Formula for energy = M1L2T-2
Here x = 1, y = 2, z = -2
In this problem
M1 = 1 kg, L1 = 1 m, T1 = 1 s and
M2 =? kg, L2 =? m, T2 =? s
So 1 Calorie = 4.2 (1/? )1 (1/? )2 (1/? )-2
= 4.
Ans.2.1
(a) The volume of a cube of side 1 is equal to
The volume of the cube = 1 cm * 1 cm * 1 cm = 1 cm3
1 cm3 = 1 * 10-6 m3
(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to1.25 * 104 (mm)2
The surface area of a solid cylinder = 2? r (r+)h, wh
8.12 A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2×1030 kg, mass of the earth = 6×1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 1011 m).
8.12 A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2×1030 kg, mass of the earth = 6×1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 1011 m).
8.12
Mass of the Sun, = kg, Mass of the Earth, = kg
Orbital radius, r = m
Let the mass of the rocket be, m

Let x be the distance from the centre of the Earth where the gravitational force acting on satellite P becomes zero.
From Newton’s law of grav
Ans.8.1
(a) No. Unlike electrical forces, gravitational force is independent of the status of the objects.
(b) Yes, the size of the space station is large enough and the astronaut will detect the change in Earth's gravity.
(c) Tidal effect depends inversely upon the cube of the distance
What is the difference between periodic and oscillatory motion?
An example of the periodic motion is Earth's revolution. In this, the motion repeats itself after regular intervals of time. In oscillatory motion, the object moves back and forth about a mean position like a vibrating spring or pendulum. It is a type of periodic motion. Although, oscillatory motion