Projectile Motion

Projectile motion is the motion of an object thrown into the air at an angle to the horizontal. The object moves in a curved path under the influence of gravity. In this scenario, a body of mass m is projected with an initial speed u at an angle θ with the horizontal.

Momentum

Momentum is a vector quantity that represents the motion of an object. It is defined as the product of mass and velocity. Mathematically, momentum (p) can be expressed as:

p = m * v

Where:
p = momentum
m = mass of the object
v = velocity of the object

Change in Momentum

The change in momentum of an object can be calculated using the following equation:

Δp = p2 - p1

Where:
Δp = change in momentum
p2 = final momentum
p1 = initial momentum

To calculate the change in momentum, we need to calculate the final momentum and initial momentum separately.

Initial Momentum

The initial momentum of the body can be calculated using the equation:

p1 = m * u

Where:
p1 = initial momentum
m = mass of the body
u = initial velocity

Final Momentum

The final momentum of the body can be calculated using the equation:

p2 = m * v

Where:
p2 = final momentum
m = mass of the body
v = final velocity

Final Velocity

To calculate the final velocity (v), we need to decompose the initial velocity (u) into its horizontal and vertical components.

The horizontal component of velocity (ux) remains constant throughout the motion and can be calculated using the equation:

ux = u * cos(θ)

The vertical component of velocity (uy) changes due to the acceleration due to gravity. The vertical motion can be analyzed using the equations of motion:

uy = u * sin(θ) - g * t

Where:
uy = vertical component of velocity
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time

The final velocity (v) can be obtained using the Pythagorean theorem:

v = √(ux^2 + uy^2)

Time

The time (T) is given in the problem statement as the duration of motion.

Calculating the Change in Momentum

Using the equations for initial momentum (p1), final momentum (p2), and time (T), we can calculate the change in momentum (Δp) as follows:

Δp = p2 - p1
Δp = m * v - m * u

Substituting the expressions for final velocity (v) and initial velocity (u), we get:

Δp = m * √(ux^2 + uy^2) - m * u

Simplifying further, we obtain the final expression for the change in momentum:

Δp = m * (√(ux^2 + uy^2) - u)

This equation gives us the change in momentum of the body after time T.