An animal-rescue plane flying due east at
40 m/s drops a bale of hay from an altitude of 73 m .
The acceleration due to gravity is
9.81 m/s.
If the bale of hay weighs 177 N , what is the
momentum of the bale the moment it strikes the ground?
part 2
At what angle of inclination will the bale strike? Answer between −180◦
and +180◦.
To find the momentum of the bale of hay the moment it strikes the ground, we can use the formula for momentum:
Momentum = mass * velocity
First, let's find the mass of the bale of hay using its weight and the acceleration due to gravity:
Weight = mass * gravity acceleration
177 N = mass * 9.81 m/s^2
mass = 177 N / 9.81 m/s^2 ≈ 18.04 kg
Now we can calculate the momentum using the mass and velocity:
Momentum = 18.04 kg * 40 m/s ≈ 721.6 kg·m/s
Therefore, the momentum of the bale of hay the moment it strikes the ground is approximately 721.6 kg·m/s.
Now, let's move on to part 2 of the question.
To find the angle of inclination at which the bale will strike, we can use trigonometry. We know the vertical distance traveled by the bale is 73 meters, and the horizontal distance traveled is determined by the time it takes the bale to reach the ground.
First, let's find the time it takes for the bale to reach the ground using the vertical distance and the acceleration due to gravity:
Vertical distance = (1/2) * gravity acceleration * time^2
73 m = (1/2) * 9.81 m/s^2 * time^2
time^2 = (73 m * 2) / 9.81 m/s^2 = 14.83 s^2
time = √14.83 s^2 ≈ 3.85 s
Now we can calculate the horizontal distance traveled by the bale:
Horizontal distance = velocity * time
Horizontal distance = 40 m/s * 3.85 s ≈ 154.0 m
The angle of inclination can be found using the tangent function:
tan(angle) = vertical distance / horizontal distance
tan(angle) = 73 m / 154.0 m
angle = arctan(73 m / 154.0 m)
Using a calculator, we find angle ≈ 26.6 degrees
However, we need the angle in the range between -180° and +180°. Since the plane is flying due east, the angle of inclination will be negative. Therefore, the angle of inclination at which the bale will strike is approximately -26.6°.
To find the momentum of the bale the moment it strikes the ground, we need to use the formula for momentum:
Momentum (p) = Mass (m) * Velocity (v)
Since we are given the weight of the bale (177 N), we can convert it to mass using the acceleration due to gravity (9.81 m/s^2):
Weight (W) = Mass (m) * Acceleration due to gravity (g)
177 N = m * 9.81 m/s^2
Solving for mass:
m = 177 N / 9.81 m/s^2
m ≈ 18.04 kg
Now we need to find the velocity of the bale when it strikes the ground. Since the plane is flying due east at 40 m/s and the bale was dropped vertically, the horizontal velocity of the bale is not relevant to its vertical motion. Thus, we can consider only the vertical motion.
We can use the kinematic equation to find the time it takes for the bale to reach the ground:
Vertical displacement (s) = Initial vertical velocity (u) * Time (t) + (1/2) * Gravity (g) * Time (t)^2
As the bale was dropped, the initial vertical velocity is 0 m/s and the vertical displacement is 73 m. Substituting these values into the equation, we can solve for the time:
73 m = 0 m/s * t + (1/2) * 9.81 m/s^2 * t^2
Simplifying the equation:
4.905 t^2 = 73 m
t^2 = 73 m / 4.905
t^2 ≈ 14.877
Taking the square root of both sides:
t ≈ √14.877
t ≈ 3.857 s
Now that we have the time it takes for the bale to fall, we can find the final vertical velocity:
Final vertical velocity (v) = Initial vertical velocity (u) + Gravity (g) * Time (t)
v = 0 m/s + 9.81 m/s^2 * 3.857 s
v ≈ 37.81 m/s
Now we can determine the momentum of the bale the moment it strikes the ground:
Momentum (p) = Mass (m) * Velocity (v)
p ≈ 18.04 kg * 37.81 m/s
p ≈ 682.19 kg*m/s
Therefore, the momentum of the bale the moment it strikes the ground is approximately 682.19 kg*m/s.
Now, let's move on to part 2, finding the angle of inclination at which the bale strikes. To do this, we can use trigonometry and the concept of vectors.
The bale was dropped vertically, so it only has a vertical velocity component. And since it is falling straight downwards, the horizontal component of its velocity remains at 0 m/s.
To find the angle, we can use the inverse tangent function (arctan) with the ratio of the vertical velocity to the horizontal velocity:
Angle (θ) = arctan (Vertical velocity / Horizontal velocity)
θ = arctan (v / 0 m/s)
However, we need to be cautious as dividing by 0 is not defined. In this case, we can say that the tan(theta) approaches infinity (since tan(theta) = sin(theta) / cos(theta)), so the angle is vertical or 90 degrees.
Since we are given the range of -180 to +180 degrees, we can express this angle as -90 or +90 degrees.
In conclusion, the angle of inclination at which the bale strikes is either -90 degrees or +90 degrees depending on the convention used for measuring angles.
h=gt²/2 => t=sqrt(2h/g) =sqrt(2•73/9.81) =3.86 s
v(y) = gt = 9.81•3.86 =37.87 m/s
v=sqrt{v(x)²+v(y)²} =sqrt{37.87²+40²}=55.08 m/s
p=mv=Pv/g =177•55.08/9.81 =993.85 kg•m/s
tan α =v(y)/v(x) =37.87/40 =0.947
α=arctan0.947 =43.43⁰