Question

An airplane is traveling at a constant horizontal speed \(v\), at an altitude \(h\) above a lake, when a trapdoor at the bottom of the airplane opens and a package is released (falls) from the plane. The airplane continues horizontally at the same altitude and velocity. Neglect air resistance. a) What is the distance between the package and the plane when the package hits the surface of the lake? b) What is the horizontal component of the velocity vector of the package when it hits the lake? c) What is the speed of the package when it hits the lake?

Short answer

Question: Calculate the distance between the package and the airplane when it hits the lake, the horizontal component of the package's velocity vector, and the speed of the package when it hits the lake, given the airplane's altitude and horizontal speed. Answer: To find the requested values, follow these steps: 1. Calculate the time it takes for the package to hit the lake using the equation: \(t = \sqrt{\frac{2h}{g}}\). 2. Find the distance between the airplane and the package when it hits the lake by multiplying the horizontal speed \(v\) with time \(t\): \(distance = vt\). 3. Determine the horizontal component of the velocity vector of the package, which is equal to the horizontal speed of the airplane: \(v_x = v\). 4. Calculate the vertical component of the package's velocity vector using the equation: \(v_y = gt\). 5. Find the speed of the package when it hits the lake using the Pythagorean theorem: \(speed = \sqrt{v_x^2 + v_y^2}\).

Step-by-step solution

Find the time it takes for the package to hit the lake

To find the time it takes for the package to hit the lake, we can use the vertical motion equation: $$h = \frac{1}{2}gt^2$$ where \(h\) is the altitude, \(g\) is the acceleration due to gravity (approximately \(9.81\,\text{m/s}^2\)), and \(t\) is the time. Solve for the time \(t\): $$t = \sqrt{\frac{2h}{g}}$$

Calculate the distance between the package and the plane when the package hits the lake

Since the airplane is traveling horizontally at a constant speed, we can find the distance between the airplane and the package when the package hits the lake by multiplying the horizontal speed \(v\) by the time \(t\) it took for the package to hit the lake: $$distance = vt$$

Calculate the horizontal component of the velocity vector of the package when it hits the lake

There is no horizontal force acting on the package once it is released from the plane. Therefore, its horizontal velocity component remains constant and equal to the horizontal speed of the airplane: $$v_x = v$$

Calculate the vertical component of the velocity vector of the package when it hits the lake

The vertical component of the velocity can be found using the equation: $$v_y = gt$$

Calculate the speed of the package when it hits the lake

Now we have both components of the velocity vector, so we can find the magnitude of the velocity (speed) using the Pythagorean theorem: $$speed = \sqrt{v_x^2 + v_y^2}$$ So the solutions for each part of the exercise are: a) The distance between the package and the plane when the package hits the surface of the lake is \(distance = vt\). b) The horizontal component of the velocity vector of the package when it hits the lake is \(v_x = v\). c) The speed of the package when it hits the lake is \(speed = \sqrt{v_x^2 + v_y^2}\).

Key concepts

Kinematics in Two Dimensions

Kinematics in two dimensions involves the study of objects moving in a plane, commonly referred to as projectile motion. When analyzing such motion, we must consider two independent components - horizontal and vertical. In our exercise, the airplane and package exhibit two-dimensional motion, with the airplane moving horizontally at a constant velocity and the package falling vertically under the influence of gravity.

Understanding the independence of the two motion components is crucial. The constant horizontal velocity of the airplane influences the horizontal movement of the package but does not affect its vertical free fall motion. Knowing this, we can separate the problem into two parts: horizontal displacement and vertical descent, which simplifies the analytic process significantly. This separation of variables is a foundational principle in two-dimensional kinematics.

Free Fall Motion

Free fall motion describes the movement of an object under the influence of gravity alone. Assuming no air resistance, this motion is characterized by a constant acceleration equal to the acceleration due to gravity, denoted by the symbol 'g', which is approximately 9.81 m/s² on Earth.

In our exercise, once the package is released, it experiences free fall and its vertical motion can be described using kinematic equations. The formula for the distance fallen, assuming initial vertical velocity is zero, is denoted by \(h = \frac{1}{2}gt^2\), where 'h' is the altitude from which the package is dropped. By solving for 't', we can determine the time it takes for the package to hit the lake's surface. This understanding of free fall motion allows us to independently solve for vertical components without interference from horizontal motion.

Constant Horizontal Velocity

Constant horizontal velocity implies that an object is moving along the horizontal axis with a velocity that does not change over time. In the context of projectile motion, this simplifies calculations, as we need not consider forces like friction or air resistance if they are negligible or absent altogether.

In the case of the airplane and its package, the horizontal component of the package's motion remains the same as the airplane's horizontal velocity after the package is released. This constant horizontal velocity allows us to determine the distance the package will travel horizontally—the same plane where the airplane continues its path—by simply multiplying the velocity by the time it takes to reach the lake. This concept highlights the ease with which problems involving constant horizontal velocities can be tackled.

Pythagorean Theorem in Velocity

The Pythagorean theorem is a mathematical principle traditionally used in geometry to relate the sides of a right triangle. In physics, particularly kinematics, we use a similar approach to find the resultant speed of an object movement in two dimensions. Speed is a scalar quantity representing the magnitude of the velocity vector.

For our exercise, the package's velocity has two components: a constant horizontal velocity (\(v_x\) equal to the airplane's speed) and an increasing vertical velocity (\(v_y\) due to free fall). To find the overall speed of the package just before it hits the water, we apply the Pythagorean theorem to these velocity components: \(speed = \sqrt{v_x^2 + v_y^2}\). This theorem helps us combine the two independent motions into a single quantifiable magnitude, concluding the full analysis of the package's trajectory upon impact.