Question

A plane, which is flying horizontally at a constant speed \(v_{\mathrm{o}}\) and at a height \(h\) above the sea, must drop a bundle of supplies to a castaway on a small raft. (a) Write down Newton's second law for the bundle as it falls from the plane, assuming you can neglect air resistance. Solve your equations to give the bundle's position in flight as a function of time \(t\). (b) How far before the raft (measured horizontally) must the pilot drop the bundle if it is to hit the raft? What is this distance if \(v_{\mathrm{o}}=50 \mathrm{m} / \mathrm{s}\) \(h=100 \mathrm{m},\) and \(g \approx 10 \mathrm{m} / \mathrm{s}^{2} ?(\mathrm{c})\) Within what interval of time \((\pm \Delta t)\) must the pilot drop the bundle if it is to land within \(\pm 10\) m of the raft?

Short answer

The bundle should be dropped 223.61 meters before the raft, with a timing precision of ±0.2 seconds.

Step-by-step solution

Apply Newton's Second Law

Newton's second law states that the net force acting on an object is equal to its mass times its acceleration. For the bundle, the only force acting on it after it's dropped is gravity. Therefore, we have \[ F = mg = ma_y \]where \( a_y = g \) is the acceleration due to gravity. This implies the vertical motion of the bundle is uniformly accelerated with acceleration \( g \).

Determine the Vertical Position as a Function of Time

For the vertical motion, we can use the kinematic equation for an object starting from rest:\[ y(t) = h - \frac{1}{2}gt^2 \]where \( y(t) \) is the vertical position at time \( t \), \( h \) is the initial height, and \( g \) is the acceleration due to gravity.

Determine the Horizontal Position as a Function of Time

In the horizontal direction, the bundle moves with constant velocity \( v_o \) because we are neglecting air resistance. Therefore, the horizontal position \( x(t) \) can be expressed as:\[ x(t) = v_o t \].

Calculate the Time to Hit the Water

The bundle hits the water when \( y(t) = 0 \). Solving \[ 0 = h - \frac{1}{2}gt^2 \] for \( t \) gives:\[ t = \sqrt{\frac{2h}{g}} \].

Determine the Horizontal Distance to Drop

The horizontal distance the bundle travels before hitting the water is the same as where the pilot must drop it to hit the raft. So,\[ x(t) = v_o \cdot \sqrt{\frac{2h}{g}} \].

Calculate Specific Values

Substitute the given values into \( x(t) \):\[ v_o = 50 \text{ m/s}, h = 100 \text{ m}, g = 10 \text{ m/s}^2 \]\[ x(t) = 50 \cdot \sqrt{\frac{2 \cdot 100}{10}} = 50 \cdot \sqrt{20} \approx 223.61 \text{ meters}. \]

Determine Time Interval for Accurate Drop

We are given that the bundle must land within \( \pm 10 \) m of the raft. Thus, we need time to satisfy:\[ v_o(t \pm \Delta t) = 213.61 \pm 10 \].Solving these equations will give the time interval:\[ \Delta x = v_o \Delta t \rightarrow \Delta t = \frac{10}{50} = 0.2 \text{ s} \]. So, the drop must occur with a timing precision of \( \pm 0.2 \) seconds.

Key concepts

Newton's Second Law

Newton's Second Law is crucial in understanding how forces affect the motion of objects. The law is formulated as \( F = ma \), where \( F \) stands for the net force, \( m \) is the mass, and \( a \) is the acceleration of the object.
This law tells us that an object will accelerate in the direction of the net force acting upon it.
In the case of the falling bundle, once it's released from the plane, the only force acting on it is gravity.

• Gravity acts vertically downwards.
• Thus, the acceleration due to gravity \( g \) is what alters the motion of the bundle.

The concept emphasizes that even if multiple forces were acting, it's the net force that determines acceleration. For the bundle, since we're ignoring air resistance, only gravity needs to be considered.

Kinematics

Kinematics is the study of motion without considering its causes. It helps describe how an object moves using quantities like displacement, velocity, and acceleration.
For the vertical motion of the bundle, kinematic equations are used:

• The equation \( y(t) = h - \frac{1}{2}gt^2 \) describes its vertical position over time.
• This shows how it falls from height \( h \) accelerating due to gravity \( g \).

For horizontal motion, which remains at constant velocity because we're neglecting air resistance, the position can be calculated as:

• \( x(t) = v_o t \) where \( v_o \) is constant horizontal velocity.

This demonstrates how the bundle’s horizontal and vertical travels are treated separately in kinematics.

Vertical Motion

Vertical motion in this context is primarily influenced by gravity. This affects how an object falls once it's in free fall.
Gravity provides a constant acceleration \( g \), approximately \( 9.8 \text{ m/s}^2 \) on Earth (rounded to \( 10 \text{ m/s}^2 \) for easier calculations).
The change in vertical position can be calculated using:

• Initial Height \( h \) from which the object falls.
• The vertical position \( y(t) = h - \frac{1}{2}gt^2 \).

The equation shows how vertical speed increases as the object descends, an example of uniformly accelerated motion.
This consistent acceleration is pivotal in predicting the fall duration before impacting the ground.

Horizontal Motion

Horizontal motion occurs alongside vertical motion, but is unaffected by gravity when air resistance is negligible.
The bundle continues at constant horizontal speed \( v_o \) due to inertia, as per Newton's First Law. Hence:

• Horizontal position over time is \( x(t) = v_o t \).

The absence of horizontal forces means there’s no horizontal acceleration.
This simplicity leads to straightforward equations for timing and distance calculations, making it possible to predict where it will land.Understanding both vertical and horizontal components enables predictions about motion's outcome jointly, despite their independent calculations.

Gravity

Gravity is a fundamental force affecting all objects with mass.It causes objects to accelerate toward the Earth at \( 9.8 \text{ m/s}^2 \), commonly approximated as \( 10 \text{ m/s}^2 \).Gravity's impact is solely vertical, influencing the bundle's descent as soon as it's released.

• It's gravity that causes the bundle to eventually hit the ground.
• It dictates the time it takes for the bundle to fall from a height \( h \).

This force is always directed toward the center of the Earth, leading to what we call 'free-fall' motion.In projectile motion, gravity is the key player that defines the vertical trajectory and timings.