An object of 3.0kg is projected into the air at a 55 degree angle. It hits the ground 3.4s later. What is it's change in momentum while it's in the air?

Y = Yo + g*Tr = 0

Yo - 9.8*(3.4/2) = 0
Yo = 16.66 m/s = Ver. component of initial velocity.

sin A = Yo/Vo
sin55 = 16.66/Vo
Vo = 16.66/sin55 = 20.34 m/s

Xo = Vo*Cos55 = 20.34*Cos55 = 11.67 m/s.

V = Xo+Yi = 11.67 + 0 = 11.67 m/s

Momentum change = M*V - M*Vo
M = 3.0 kg
V = 11.67 m/s
Vo = 20.34 m/s

To calculate the change in momentum while the object is in the air, we need to find the initial and final momentum of the object.

Step 1: Find the initial momentum:
The initial momentum (p_initial) is given by the formula:
p_initial = mass × velocity_initial

Given:
Mass (m) = 3.0 kg

To find the initial velocity (v_initial), we can use the horizontal and vertical components of the velocity.

Step 2: Find the horizontal component of the initial velocity:
The horizontal component of the initial velocity remains constant throughout the motion. It can be found using the formula:
v_initial_horizontal = velocity_initial × cos(angle)

Given:
Angle = 55 degrees

Using trigonometric functions:
cos(angle) = cos(55)

Using a scientific calculator:
cos(55) ≈ 0.5736

Step 3: Find the vertical component of the initial velocity:
The vertical component of the initial velocity changes due to the gravitational acceleration acting on the object. It can be found using the formula:
v_initial_vertical = velocity_initial × sin(angle)

Given:
Angle = 55 degrees

Using trigonometric functions:
sin(angle) = sin(55)

Using a scientific calculator:
sin(55) ≈ 0.8192

Step 4: Find the magnitude of the initial velocity:
The magnitude of the initial velocity (v_initial_magnitude) can be found using the Pythagorean theorem:
v_initial_magnitude = sqrt(v_initial_horizontal^2 + v_initial_vertical^2)

Using the values from the previous steps:
v_initial_magnitude = sqrt(v_initial_horizontal^2 + v_initial_vertical^2)
v_initial_magnitude = sqrt((v_initial × cos(angle))^2 + (v_initial × sin(angle))^2)
v_initial_magnitude = sqrt((v_initial^2 × cos^2(angle)) + (v_initial^2 × sin^2(angle)))
v_initial_magnitude = sqrt(v_initial^2 × (cos^2(angle) + sin^2(angle)))
v_initial_magnitude = sqrt(v_initial^2)

Therefore, v_initial_magnitude = v_initial

Since the object is projected into the air, it starts with an initial vertical velocity only and no initial horizontal velocity. Hence, the initial horizontal component of the velocity (v_initial_horizontal) is 0.

v_initial_horizontal = 0 m/s

Step 5: Calculate the initial vertical velocity:
We are given that the object hits the ground 3.4 seconds later. The object was in the air for this time period, so using the equation of motion for vertical motion:

v_final_vertical = v_initial_vertical + (acceleration × time)

Given:
Acceleration due to gravity (g) ≈ 9.8 m/s²
Time (t) = 3.4 s

Substituting the known values:
0 = v_initial_vertical + (9.8 × 3.4)

Simplifying the equation:
0 = v_initial_vertical + 33.32

Rearranging the equation:
v_initial_vertical = -33.32 m/s

The negative sign indicates that the initial vertical velocity is in the opposite direction of the gravitational acceleration.

Step 6: Calculate the initial magnitude of the velocity:
Using the initial horizontal and vertical components of velocity, we can calculate the magnitude of the initial velocity (v_initial_magnitude) using the Pythagorean theorem:

v_initial_magnitude = sqrt(v_initial_horizontal^2 + v_initial_vertical^2)
v_initial_magnitude = sqrt(0^2 + (-33.32)^2)
v_initial_magnitude = sqrt(0 + 1107.7024)
v_initial_magnitude = sqrt(1107.7024)

Using a scientific calculator:
sqrt(1107.7024) ≈ 33.30 m/s

Therefore, the magnitude of the initial velocity (v_initial_magnitude) is approximately 33.30 m/s.

Step 7: Calculate the initial momentum:
The initial momentum (p_initial) can now be calculated using the formula:
p_initial = mass × velocity_initial

Given:
Mass (m) = 3.0 kg
Velocity_initial = v_initial_magnitude = 33.30 m/s

Substituting the known values:
p_initial = 3.0 kg × 33.30 m/s
p_initial = 99.90 kg·m/s

Therefore, the initial momentum (p_initial) of the object is approximately 99.90 kg·m/s.

To find the change in momentum of an object while it's in the air, we need to know its initial momentum and its final momentum. The initial momentum can be calculated using the formula:

Initial Momentum = Mass × Initial Velocity

The final momentum can be calculated using the formula:

Final Momentum = Mass × Final Velocity

Since the object is projected into the air, we can find its initial velocity using the given angle. The horizontal component of the initial velocity can be found using the formula:

Initial Velocity (horizontal component) = Initial Velocity × cos(angle)

The vertical component of the initial velocity can be found using the formula:

Initial Velocity (vertical component) = Initial Velocity × sin(angle)

Since the object hits the ground, its final velocity in the vertical direction would be zero. Therefore, we only need to find the horizontal component of the final velocity.

To find the initial velocity, we need to use the given information of the angle and the time taken to hit the ground.

Using the formula of horizontal motion:

Horizontal Distance = Initial Velocity (horizontal component) × Time

We can rearrange the formula to solve for Initial Velocity (horizontal component):

Initial Velocity (horizontal component) = Horizontal Distance / Time

Now, we can substitute the given values into the equation:

Horizontal Distance = 3.4s (since the object hits the ground 3.4 seconds later)
Time = 3.4s
Initial Velocity (horizontal component) = ?

Substituting the values in, we get:

Initial Velocity (horizontal component) = Horizontal Distance / Time
Initial Velocity (horizontal component) = 3.4s / 3.4s
Initial Velocity (horizontal component) = 1m/s

Since the object is projected at an angle of 55 degrees, we can use trigonometry to find the initial velocity. We already have the initial velocity's horizontal component, and we can find the initial velocity's vertical component using the formula:

Initial Velocity (vertical component) = Initial Velocity (horizontal component) / tan(angle)

Now, substituting the values into the equation:

Initial Velocity (vertical component) = 1m/s / tan(55°)
Initial Velocity (vertical component) ≈ 0.731m/s

Now, we can calculate the initial momentum:

Initial Momentum = Mass × Initial Velocity
Initial Momentum = 3.0kg × 0.731m/s
Initial Momentum ≈ 2.193 kg·m/s

Since the object hits the ground and has a final velocity of zero in the vertical direction, we only need to calculate the final velocity in the horizontal direction.

To find the final velocity in the horizontal direction, we can use the formula:

Horizontal Distance = Final Velocity (horizontal component) × Time

Rearranging the formula to solve for Final Velocity (horizontal component):

Final Velocity (horizontal component) = Horizontal Distance / Time

Substituting the given values:

Horizontal Distance = 3.4s
Time = 3.4s
Final Velocity (horizontal component) = ?

Substituting the values in, we get:

Final Velocity (horizontal component) = Horizontal Distance / Time
Final Velocity (horizontal component) = 3.4s / 3.4s
Final Velocity (horizontal component) = 1m/s

Finally, we can calculate the final momentum:

Final Momentum = Mass × Final Velocity
Final Momentum = 3.0kg × 1m/s
Final Momentum = 3.0 kg·m/s

The change in momentum while the object is in the air is the difference between the final momentum and the initial momentum:

Change in Momentum = Final Momentum - Initial Momentum
Change in Momentum = 3.0 kg·m/s - 2.193 kg·m/s
Change in Momentum ≈ 0.807 kg·m/s

Therefore, the change in momentum while the object is in the air is approximately 0.807 kg·m/s.