Multiple Choice Questions (MCQs)

2.1 The angle at which dot product becomes equal to cross product is:
(a) 65°
(b) 45° ✓
(c) 76°
(d) 30°
✅ Explanation:
Dot product: A⋅B=ABcos
Cross product: A×B=ABsin
If cosθ=sinθ , then = sin θ=45∘

2.2 The projectile gains its maximum height at an angle of:
(a) 0°
(b) 45° ✓
(c) 60°
(d) 90°
✅ Explanation:
Maximum height is part of maximum range condition, which is at 45°.

2.3 The scalar product of two vectors is maximum if they are:
(a) perpendicular
(b) parallel ✓
(c) at 30°
(d) at 45°
✅ Explanation:
Scalar (dot) product is maximum when cosθ=1 i.e., θ=0∘ vectors are parallel.

2.4 The range of projectile is same for two angles which are mutually:
(a) perpendicular
(b) supplementary
(c) complementary ✓
(d) 270°
✅ Explanation:
Range R=v2sin(2θ) / g

So, if θ1+θ2=90∘ then sin(2θ1)=sin(2θ2) → complementary angles.

2.5 The acceleration at the top of a trajectory of projectile is:
(a) maximum
(b) minimum
(c) zero
(d) g ✓
✅ Explanation:
Acceleration due to gravity acts constantly downward with magnitude ggg, even at the top.

2.6 SI unit of impulse is:
(a) kg·m/s²
(b) N·m

(c) N·s ✓
(d) N·m/s
✅ Explanation:
Impulse = Force × Time = N × s = N·s

2.7 The rate of change of momentum is:
(a) force ✓
(b) impulse
(c) acceleration
(d) power
✅ Explanation:
F=dp / dt⇒Force is the rate of change of momentum

2.8 As rocket moves upward during its journey, then its acceleration goes on:
(a) increasing ✓
(b) decreasing
(c) remains same
(d) moves with uniform velocity
✅ Explanation:
As fuel burns, mass decreases while thrust remains, so acceleration increases.

2.9 Elastic collision involves:
(a) loss of energy
(b) gain of energy
(c) no gain, no loss of energy ✓
(d) no relation between energy and elastic collision
✅ Explanation:
In an elastic collision, both momentum and kinetic energy are conserved.

Short Answer Questions

2.1 – State Right Hand Rule for Two Vectors (Vector Product)
 Answer:
The Right-Hand Rule is used to determine the direction of the vector product (cross product)
of two vectors.
1- Point the fingers of your right hand in the direction of the first vector (A).
2- Rotate them toward the second vector (B) through the smallest angle.
3- Your thumb will point in the direction of the resultant vector (A × B).
✳️ Cross product is perpendicular to both vectors.

2.2 – Define Impulse and Show How It is Related to Momentum
Answer:
Impulse:
Impulse is the product of force and the time interval during which the force acts.
Impulse=F⋅Δt
Relation with Momentum:
According to Newton’s second law:
F=Δp / Δt⇒F⋅Δt=Δp
✅ So,
Impulse=Change in Momentum

2.3 – Differentiate Between Elastic and Inelastic Collision
 Answer:
Property Elastic Collision Inelastic Collision
Kinetic Energy Conserved Not Conserved
Momentum Conserved Conserved
Example Billiard balls Car crash, clay hitting wall

2.4 – Show That Rate of Change in Momentum is Equal to Force & State
Newton’s 2nd Law
 Answer:
From Newton’s Second Law:
F=dp/dt
Where:
 F = Force
 p=mv = Momentum
This means:
✅ Force is equal to the rate of change of momentum.
Newton’s Second Law (in terms of momentum):
“The force acting on a body is equal to the rate of change of momentum produced in the body.”

2.5 – State Law of Conservation of Linear Momentum and Condition for Its Validity.

 Answer:
Law of Conservation of Linear Momentum:
“The total linear momentum of an isolated system remains constant, if no external force acts on
it.”
Initial Momentum=Final Momentum
✅ Condition:
This law holds only when the system is closed and isolated — i.e., no external force is acting
on the system.
2.6 – Show that Range of Projectile is Maximum at 45°
 Answer:
Post-Collision Motion Objects may rebound Objects may stick together

The formula for range:
R=u^2sin(2θ)/ g
✅ Range depends on sin(2θ) which is maximum when sin(2θ)=1
This happens when:
2θ=90∘⇒θ=45∘
Therefore, the range of a projectile is maximum at an angle of 45°.

2.7 – Find Time of Flight to Reach Maximum Height
Answer:
Time to reach maximum height is half of total time of flight:
T=usinθ / g
Where:
 u = initial velocity
 θ = angle of projection
 g = acceleration due to gravity
✅ Time to reach max height = T=usinθ / g

2.8 – Max Range is 800 m, Find Height at 60°
 Given:
 Rmax=800 m
 Angle θ=60∘
� Use the formula:
H=Rmax / 4 . tanθ
H=800/ 4 ⋅tan(60∘)=200 ⋅ √3≈346.4 m

✅ Height attained ≈ 346.4 m

 Constructed Response Questions

2.1 – Why Does a Hunter Miss the Bird When Aiming Directly at It?
 Answer:
Because of gravity, the bullet or projectile follows a curved path, while the bird may fly away
or stay still.
✅ So, aiming directly results in the bullet falling below the target.
That’s why hunters aim slightly above the bird.

2.2 – Why Does a Person Fall Safely on Sand, but Not on Concrete?
Answer:
Sand increases the time of impact, reducing the rate of momentum change, hence reducing
force (as per impulse-momentum theorem).
F=Δp / ΔtF
✅ Sand gives more time, so less force is felt.

2.3 – Conditions for Birds to Fly in Air
Answer:
Birds fly due to Newton’s 3rd law:
 They push air downward with their wings.
 The air gives an equal and opposite upward lift.
 The lift force must balance the bird’s weight for steady flight.

2.4 – Describe Situations with v = 0, a = 0, etc.

 Answer:
1. v=0 but a≠0
A ball at the top of its projectile path — velocity momentarily zero, but gravity is acting.
2. a=0 , but v≠0
A vehicle moving at constant speed in a straight line — no acceleration.
3. v⊥a:
In uniform circular motion, velocity is tangential and acceleration (centripetal) is
toward the center → perpendicular.

2.5 – Effect of Air Resistance on Range of Projectile
Answer:
Air resistance:
 Reduces the horizontal component of velocity.
 Decreases the total range.
 Makes trajectory asymmetric — descent is steeper.
✅ Actual range is less than theoretical range (without air).

Comprehensive Questions

2.1 – Define and Explain Scalar Product. Write Its Characteristics
Answer:
Scalar Product (Dot Product):
The scalar (or dot) product of two vectors A and B is given by:
A ⋅B=ABcosθ
Where:
 A and B are magnitudes of vectors
 θ is the angle between them
 Result is a scalar quantity

 Characteristics:

1. A ⋅B=B⋅A → Commutative
2. A⋅B=0 if vectors are perpendicular
3. A⋅A=∣A∣2
4. Result is maximum when θ=0∘

2.2 – Define and Explain Vector Product. Characteristics of Vector Product
Answer:
Vector Product (Cross Product):
A×B=Absinθ n^
Where n^ is a unit vector perpendicular to both A and B (right-hand rule).

Characteristics:
1. A×B=−B×A → Anti-commutative
2. A×A=0
3. Result is a vector perpendicular to plane of A and B
4. Maximum when angle is 90∘

2.3 – Derive Three Equations of Motion (Graphical Method)
Answer:
Using velocity-time graph:
1. First Equation:
v=ui+at
2. Second Equation:
Displacement = Area under v–t graph
s=ut+ 1 / 2at^2
3. Third Equation:

Eliminate time ttt:
V^2=u^2+2as

2.4 – What is Projectile Motion? Explain.
 Answer:
Projectile Motion:
The curved path followed by an object thrown near Earth’s surface under gravity alone is called
projectile motion.
 Horizontal velocity is constant
 Vertical motion is like free fall
 Path is a parabola

2.5 – Derive Expressions for Projectile Motion
(i) Time of Flight
T=2usinθ / g
(ii) Maximum Height
H=u^2sin^2θ / 2g
(iii) Range
R=u^2sin(2θ) / g

2.6 – Explain Elastic Collision in 1D & Relative Velocities
 Answer:
Elastic Collision (1D):
Both momentum and kinetic energy are conserved.
 Condition:
m1u1+m2u2=m1v1+m2v2(momentum)

1 / 2 m1u1^2+ 1 / 2 m2u2^2= 1 / 2 m1v1^2+ 1 / 2 m2v2^2(KE
✅ Relative Velocity Before = After (Reversed):
u1−u2=−(v1−v2)

2.7 – Derive Momentum & Energy Conservation in 2D Collision
 Answer:
Let two particles collide with masses m1m_1m1 and m2m_2m2, and split motion into x and y
components:
 Momentum in x-direction:
m1u1x+m2u2x=m1v1x+m2v2x
 Momentum in y-direction:
m1u1y+m2u2y=m1v1y+m2v2y
 Kinetic Energy (Elastic Collision Only):
1 / 2m1u1^2+1 / 2m2u2^2=1 / 2m1v1^2+1 / 2m2v2^2

2.8 – Explain Inelastic Collision in Two Dimensions
 Answer:
 In inelastic collisions, momentum is conserved, but kinetic energy is not.
 Bodies may stick together or move separately with energy loss (sound, heat,
deformation).
Apply conservation of momentum in both axes:
 m1u1x+m2u2x=(m1+m2)vx
 m1u1y+m2u2y=(m1+m2)vy
✅ KE is not conserved, unlike elastic collisions.