The initial vertical velocity viy= 0m/s Now vy= viy+g.t =0+9.81m/s. 2s =19.62m/s
Rounding to two significant figures, this is approximately
20m/s

They are independent of eachother throughout projectile motion

When a stone is thrown horizontally from the top of a tall building, it experiences only the force of gravity acting vertically downward. Since there is no initial vertical velocity, the stone’s motion is governed solely by the downward force of gravity. The path of the stone is a result of the combination of its horizontal velocity (due to the initial throw) and the vertical acceleration due to gravity. As the stone falls vertically, it also continues to move horizontally due to its initial horizontal velocity. The resulting trajectory of the stone is a curved path known as a parabola.

In case of projectile motion the vertical component of particle’s velocity changes continuously because of the force acting in vertical direction which is its own weight ($mg$).

But in the horizontal direction as there is no force acting on the
object ; its horizontal velocity remains constant

During projectile motion, the vertical component of velocity:

• Decreases as the projectile rises (due to gravity acting downwards).
• Becomes zero at the peak of the trajectory.
• Increases (in the negative direction) as the projectile falls back down.

The ratio of the ranges for two complementary angles in projectile motion is equal to one because the Range is the same for both angles.

at complementary angles we have same Range values

Explanation:

At the top of the flight, the vertical speed becomes zero but the horizontal speed stays.
If initial kinetic energy is K,
then K = ½ m v².
At the highest point,
speed = v cos 60°.
So KE_top = ½ m (v cos 60°)²
= ½ m v² cos² 60°
= K × (1/2)² = K/4

Explanation:

💡 Concept:

For maximum range, the angle of projection is 45°.

At the highest point of the projectile:

• The vertical velocity component becomes zero.

• The horizontal velocity remains constant throughout the motion (no acceleration in horizontal direction assuming no air resistance).

Horizontal component of velocity:

At highest point, only horizontal component exists.

Explanation:

Formula for Maximum Range:

Explanation:

as we know at highest point the vertical component of velocity is zero

Explanation

• Constant downward acceleration (due to gravity) implies the vertical velocity changes at a constant rate.
• A constant rate of change in velocity (i.e., constant acceleration) produces a straight line on a velocity–time graph.
• The slope is negative because gravity acts downward, reducing upward velocity or increasing downward velocity.

Explanation 

• A projectile is any object upon which the only force is gravity,
• Projectiles travel with a parabolic trajectory due to the influence of gravity,
• There are no horizontal forces acting upon projectiles and thus no horizontal acceleration,
• The horizontal velocity of a projectile is constant (a never changing in value),
• There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down,
• The vertical velocity of a projectile changes by 9.8 m/s each second,

Explanation

Mass is not relevant in projectile motion. What is relevant is the initial speed and direction of the object.

Explanation:

The Acceleration due to Gravity is 9.8 m/Sec^2

The maximum height of a object in a projectile trajectory occurs when the vertical component of velocity, Vy , equals zero. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decelerate.

EXPLANATION:

V₀x = V₀ * cos(θ)

Where:

• V₀x is the horizontal component of the initial velocity.
• V₀ is the initial velocity.
• θ is the launch angle 

V₀x = 3 * cos (60)
= 3 m/s * 0.5
= 1.5 m/s

EXPLANATION:

In his book Two New Sciences, Galileo stated that for projectile angles that exceed or fall short of 45° by equal amounts, the ranges are equal.

This means that if a projectile is launched at two angles that are complementary (e.g., 30° and 60°), the horizontal ranges will be the same, assuming the same initial speed and ideal conditions (no air resistance).

Explanation

The velocity is perpendicular to the acceleration due to gravity at the highest point. So, the momentum is also perpendicular to the acceleration due to gravity at the highest point.

Explanation

Explanation

The path of the projectile with respect to another projectile will be a straight line as the acceleration of both the projectiles will be same. So, they will have only a relative velocity with respect to each other.

Explanation

The acceleration of the projectile is always equal to g in downward direction during the flight.

Explanation

The velocity of the bomb w.r.to the pilot, in horizontal direction = 0.

So, the pilot will see the bomb falling vertically downward as the bomb do not have any horizontal motion and only vertical motion w.r.to the pilot.

Explanation

Explanation

From the figure, the X-component remains unchanged, while the Y-component is reversed. Then, the velocity at point B is (2i-3j) ms^-1

Audio Explanation: 

Explanation

Since horizontal component of velocity is constant, hence momentum is constant

Explanation

Only horizontal component of velocity $\left(u\cos \theta \right)$

Explanation

It is clear that range is proportional to the direction (angle) and the initial speed.

Explanation

Explanation

Acceleration through out the projectile motion remains constant and equal to g

Explanation

Explanation

Explanation

Explanation

$R=4H\mathrm{cot\theta }$

if $\theta =45°$ then $R=4H\cot (45°)=4H$

Explanation

when $\theta =90°$, R = 0 i.e. the body will fall at the point of projection after completing one dimensional motion under gravity.

Explanation

An external force by gravity is present throughout the motion so momentum will not be conserved.

Explanation

. If initial velocity be doubled then range will become four times.

Explanation

Due to air resistance, it’s horizontal velocity will decrease so it will fall behind the aeroplane.

Explanation

Due to constant velocity along horizontal and vertical downward force of gravity stone will hit the ground following parabolic path

Explanation

Bullet will take s=vt=> t=s/v $\frac{\mathrm{100/}}{1000}=0.1$sec to reach target

During this period vertical distance (downward) travelled by the bullet

So the gun should be aimed 5 cm above the target.

The only change in velocity due to gravity acts vertically downwards. This change is independent of the initial velocity and angle of projection and is given by the acceleration due to gravity g multiplied by time t.

Explanation

Explanation:

In projectile thrown at angle θ Range R and maximum height H are given as :
Rtanθ=4H
now
R=H
tanθ=4H/R
tanθ=4H/H
tanθ=4
θ=tan¯¹(4)
θ=76°

A projectile such as a ball thrown undergoing a parabolic path is made of two kinds of motion. They are the horizontal motion and the vertical motion. The horizontal motion is a uniform motion. It has no acceleration while the vertical motion is an accelerated motion. The direction of the acceleration is always towards the ground.

2gh = Vf²Vi²
h= V²/2g
h= (15)²/2g
h= 11m

Acceleration of the ball during any stage is constant in g and direction ,it is g=9.8m/s^2
and its downwards.

Two angles are called complementary when their measures add to 90 degrees. Two angles are called supplementary when their measures add up to 180 degrees.

When the stones were thrown from the top of a tall building due to the presence of gravity it follows the projectile motion. The projectile motion follows the trajectory of a parabolic path throughout the journey.

Explanation:

Both projectiles are only under the influence of gravity.
So, each has the same acceleration downward, which is 9.8 m/s².
Relative acceleration = acceleration of one – acceleration of other
= 9.8 m/s² – 9.8 m/s² = 0

Short Trick:
If both are in free fall or projectile motion under gravity,
relative acceleration is always zero.

Explanation:

At the highest point of a projectile’s path, it stops rising and is about to fall.
So, the vertical component of velocity becomes zero at that instant.
Only the horizontal component remains, which stays constant throughout the motion.

Explanation:

Since the acceleration due to gravity is vertically downward, at the highest point Q, where the direction of motion is to the right (horizontal velocity remains constant since air resistance, which opposes motion, is negligible [also, it is not affected by the vertically down acceleration], so the horizontal component of acceleration is zero).

At the highest point Q, there is no vertical upward component since the projectile has reached its maximum vertical height – it cannot go any higher.

The horizontal component of the projectile’s velocity is not zero since the motion continues towards the right (note that zero acceleration does not mean a speed of zero – it just means that the speed is not changing). [B is incorrect]

Both the kinetic energy and momentum of the projectile depends on its speed. Since the speed is not zero, the kinetic energy and momentum cannot be zero. [C and D are incorrect]

Explanation:

In the motion of a projectile (ignoring air resistance):

• The horizontal component of velocity remains constant because there is no force acting horizontally.

• The acceleration due to gravity (9.8 m/s²) acts vertically downward throughout the flight and remains constant.

In contrast:

• The vertical component of velocity changes continuously due to gravity.

• Kinetic energy and potential energy also change at different points in the path.

Therefore, the correct pair of physical quantities that stay constant during projectile motion is:

Horizontal velocity and acceleration due to gravity

Hint: An External Force by gravity is Present Throughout the Motion so Momentum cannot Be Conserved

Explanation:

When a projectile moves freely under gravity (and air resistance is neglected), the mechanical energy (i.e., the sum of kinetic and potential energy) remains constant throughout the motion.

• At the highest point, potential energy is maximum, and kinetic energy is minimum.

• At the lowest point, kinetic energy is maximum, and potential energy is minimum.

• But the total energy remains the same at all points.

However, momentum is not conserved during the flight because:

• The gravitational force acts on the projectile, changing its momentum over time.

• Momentum is only conserved in a system with no external forces, and gravity is an external force in this case.

Hint:
Hence Velocity is Horizontal and Acceleration is Vertical

Explanation:

At the top of a projectile’s trajectory:

• The vertical component of velocity becomes zero.

• The horizontal component of velocity remains (since there’s no force acting horizontally).

• The velocity is thus entirely horizontal at that point.

However, the acceleration due to gravity always acts vertically downward throughout the projectile’s motion, including at the top.

So, at the highest point:

• Velocity is horizontal

• Acceleration is vertical downward

That makes the angle between velocity and acceleration = 90°

Hint:  
and R both are proportional to u² Hence, percentage increase in horizontal range would also be 10%

Explanation:

If the maximum height increases by 10%, it means 𝑢² increased by 10%. Since range is also proportional to 𝑢², it will increase by 10% as well.

Explanation:

Explanation:

In oblique projectile motion:

• The vertical component of velocity decreases due to gravity, becomes zero at the highest point, then increases downward.

• The horizontal component remains constant throughout.

• The net velocity (combination of vertical and horizontal) decreases, reaches minimum at the top (when vertical velocity is zero), and then increases again.

Explanation:

When a particle moves with uniform velocity, it means both the speed and direction remain constant, so:

• Speed stays constant

• Velocity does not change

• Acceleration is zero

However, the position vector (which tells the location of the particle with respect to origin) keeps changing as the object moves over time.

Explanation

  g points down in projectile motion. Always

As it is a about a projectile motion, the kinetic energy obviously minimum at the greatest height of the motion, but not 0. Because at the greatest height of the motion although the V(y)=0, there still exist the constant V(x) which is counted for kinetic energy.

So at the greatest height of a projectile motion, the kinetic energy of a body is minimum and the equation for this value is
K.E.= 1/2×(mv²)

Two projectiles fired with the same speed but at different angles can have the same range if the sum of their angles of projection is 90 degrees.

Standard unit vectors follow i × j = k.
Here A × B = (2i) × (3j)
= 6 (i × j).
So A × B = 6k.

Explanation:

Explanation:

For parallel vectors, the angle between them is 0°.
Sine of 0° is zero.
So |A × B| = |A||B| sin0° = 0, meaning the cross product is the zero vector.

Explanation:

B is 2·A (parallel to A), so A × B = 0 for any parallel vectors.

Cross product is defined to give a new vector quantity.
This new vector is perpendicular to the plane containing the original two vectors.
So the result of A × B is always a vector, not a scalar.

For vectors A and B with angle θ between them, the cross product magnitude is given by |A × B|.
By definition, |A × B| = |A||B| sinθ.
So the magnitude depends on both magnitudes and sine of the included angle.

Explanation:

Explanation:

Cross product is zero if vectors are parallel (θ = 0° or 180°) or if either vector is a zero vector.
Because sin0° = sin180° = 0.

Vector A and λA have the same or opposite direction depending on sign of λ.
So they are parallel or antiparallel vectors.
Cross product of parallel vectors is zero, hence A × (λA) is zero vector.

Cross product magnitude is |A × B| = |A||B| sinθ.
If A and B are non-zero and A × B = 0, then sinθ must be zero.
Sine becomes zero at θ = 0° or 180°, meaning the vectors are parallel or antiparallel.

Cross product produces a vector that is normal to the plane of the two vectors.
This new vector is at right angle to both A and B.
So the direction of A × B is perpendicular to the plane containing A and B.

Magnitude of cross product is |A × B| = |A||B| sinθ.
This is maximum when sinθ is maximum.
Sine has maximum value 1 at θ = 90°, so cross product is maximum at 90°.

Torque magnitude is |τ| = |r × F|.
By cross product definition, |r × F| = r F sinθ.
Here θ is the angle between r and F.

Explanation:

|A × B| = |A| |B| sin θ
A · B = |A| |B| cos θ

Given:
|A| |B| sin θ = √3 (|A| |B| cos θ)
⇒ sin θ = √3 cos θ
⇒ tan θ = √3
⇒ θ = 60°

Explanation:

The vector (B × A) is perpendicular to both B and A.
So when we take the **dot product** of (B × A) with A, the result is:

(B × A) · A = 0

Reason: dot product of two perpendicular vectors is zero. 

We know that i × j = k in right-handed system.
Reversing the order gives j × i = −(i × j).
So j × i = −k.

From cross product magnitude, |A × B| = |A||B| sinθ.
Given |A × B| equals |A||B|, so sinθ must be 1.
Sine becomes 1 at θ = 90°, so the vectors are perpendicular.

The vector r × F represents torque or moment of force.
Force has unit newton and position vector has unit metre.
So unit of torque is newton metre (N m).

Associative law would require (A × B) × C to equal A × (B × C).
For cross product these two expressions usually give different vectors.
Therefore vector product does not satisfy associative law.

Magnitude |A × B| = |A||B| sinθ depends on sine of angle between them.
It is maximum when sinθ is maximum.
Sine function reaches its maximum value 1 at 90°, so perpendicular vectors give largest |A × B|.

Magnitude |A × B| gives area of parallelogram formed by A and B.
A triangle formed by the same two vectors has half of this area.
So ½|A × B| represents area of the triangle formed by A and B.

Explanation:

|A × B| = |A| |B| sin θ
= 5 × 7 × sin 90°
= 35.

Explanation:

A × B = –(B × A)
So, the two vectors are equal in magnitude but opposite in direction,
hence the angle between them is π radians (180°).

Magnitude of vector product depends on sine of angle between vectors.
By definition, |A × B| = |A||B| sinθ.
So the cross product magnitude is proportional to sine of the included angle.

Explanation:

Explanation:

Calculate cross product using determinant:

Magnitude |A × B| equals |A||B| sinθ.
This is equal to the area of a parallelogram having sides A and B.
So cross product magnitude represents area of the parallelogram formed by two vectors.

Explanation:

A · B = 3×6 + 4×8
= 18 + 32 = 50 ❌ (not 48)

Vector product in components is written using a 3 × 3 determinant.
First row contains unit vectors i, j, k.
Second and third rows contain components of A and B respectively.
Expanding this determinant gives the components of A × B.

If A, B and C are coplanar, they all lie in the same plane.
Vector A × B is perpendicular to the plane containing A and B.
Since C is also in that plane, A × B is perpendicular to C as well.

Explanation:

The direction of the cross product vector is given by the right-hand rule: if fingers point along A and curl towards B, then the thumb points in the direction of A × B.

In the right-hand rule, you first point the fingers of your right hand along A.
Then curl the fingers towards B through the smaller angle between them.
Your extended thumb then points in the direction of A × B.

Explanation:

Its magnitude = √(6² + 4²)
= √(36 + 16) = √52

Take A, B and C as i, j and k in a right-handed coordinate system.
We know that i × j = k and k is a unit vector.
Dot product of k with itself is 1.
So (A × B) · C equals 1.

Explanation:

The cross product A × B produces a vector that is perpendicular to both A and B following the right-hand rule. Its magnitude equals |A||B|sinθ, where θ is the angle between A and B.

Explanation:

Using right-hand rule / cyclic order î → ĵ → k̂, we have î × k̂ = –(k̂ × î) = –ĵ.

Cross product of two vectors should give a vector quantity.
Dot product gives a scalar result.
If the answer is scalar while calculating cross product, it means dot product formula was used by mistake.

Cross product satisfies A × (B + C) = A × B + A × C, so distributive law holds.
It also obeys (kA) × B = k(A × B), so scalar multiplication behaves well.
But it is not commutative, because B × A = −(A × B).
So the commutative law does not hold for vector product.

The cross product obeys distributive law over addition.
So A × (B + C) expands as A × B plus A × C.
Therefore A × (B + C) = A × B + A × C.

Swapping the order in a cross product changes the sign of the result.
So B × A equals −(A × B).
This property is called anti-commutativity of the vector product.

Torque measures turning effect of a force about a point.
By definition, torque is given by cross product of position vector r and force F.
So τ = r × F.

Compute A × B = (3i + 4j) × i.
This equals 3(i × i) + 4(j × i).
i × i is zero and j × i = −k.
So A × B = 0 + 4(−k) = −4k.

Cross product magnitude is |A × B| = |A||B| sinθ.
If |A × B| is zero, then either |A| or |B| is zero or sinθ is zero.
Sinθ is zero when vectors are parallel or antiparallel.
So cross product is zero if one vector is zero or the vectors are parallel.

Magnetic force on a moving charge is given by F = q v × B.
Here v is velocity vector and B is magnetic field vector.
This force is a vector product of charge velocity and magnetic field.

Explanation: