An object of mass 4.3 kg is projected into the air at a 45° angle. It hits the ground 3.8 s later. What is its change in momentum while it is in the air? Ignore air resistance.
To find the change in momentum of the object while it is in the air, we first need to calculate its initial momentum and final momentum.
The momentum of an object is given by the equation P = m * v, where P is the momentum, m is the mass of the object, and v is its velocity.
Given that the mass of the object is 4.3 kg, we need to calculate its initial and final velocities.
To find the initial velocity, we can use the fact that the object is projected at a 45° angle. In a projectile motion, the initial velocity can be split into horizontal and vertical components. The horizontal component remains constant throughout the motion, while the vertical component changes due to gravity.
Since the object hits the ground 3.8 s later, we can use the equation d = v0 * t + 0.5 * a * t^2, where d is the vertical distance covered, v0 is the initial vertical velocity, t is the time, and a is the acceleration due to gravity (-9.8 m/s^2).
Since the object is projected at a 45° angle, the initial vertical velocity is v0 * sin(45°), and the initial horizontal velocity is v0 * cos(45°). The vertical distance covered is given by d = -0.5 * a * t^2 because the final position is taken to be the origin.
Plugging in the values, we have:
-0.5 * (-9.8 m/s^2) * (3.8 s)^2 = v0 * sin(45°) * (3.8 s)
Solving for v0, we get:
v0 = 3.8 s * (-0.5 * (-9.8 m/s^2) * (3.8 s)^2) / (sin(45°))
The initial horizontal velocity can be found by multiplying v0 by cos(45°).
Now that we have the initial velocity, we can calculate the initial momentum:
P_initial = m * (v_horizontal_initial + v_vertical_initial)
Next, we need to find the final velocity. Since we are ignoring air resistance, the only force acting on the object is gravity. The final vertical velocity can be found using the equation v_final = v0 + a * t, where a is the acceleration due to gravity (-9.8 m/s^2) and t is the time (3.8 s).
The final horizontal velocity remains the same as the initial horizontal velocity because there is no horizontal force acting on the object.
Now we can calculate the final momentum:
P_final = m * (v_horizontal_final + v_vertical_final)
Finally, the change in momentum is given by the difference between the final momentum and the initial momentum:
Change in momentum = P_final - P_initial
By following these steps, you can calculate the change in momentum of the object while it is in the air.