Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by
x = 100t, y = −4.9t² + 4000, t ≥ 0
where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

Verified step by step guidance

1. Identify the variables and given information: The horizontal position is given by \(x = 100t\), and the vertical position is \(y = -4.9t^{2} + 4000\). The target is on the ground, so the vertical position at impact is \(y = 0\).
2. Set the vertical position equation equal to zero to find the time \(t\) when the packet hits the ground: \(-4.9t^{2} + 4000 = 0\).
3. Solve the quadratic equation for \(t\) to find the time of impact. Since \(t \geq 0\), take the positive root of the equation.
4. Use the time of impact \(t\) in the horizontal position equation \(x = 100t\) to find the horizontal distance traveled by the packet from the release point to the target.
5. The horizontal distance found is how far before the target the packet must be released, so the plane should release the packet this many meters before reaching the target.

Key Concepts

Parametric Equations of Motion

Parametric equations express the position of an object as functions of time, separating horizontal and vertical components. Here, x = 100t represents horizontal motion at constant speed, while y = -4.9t² + 4000 models vertical motion under gravity. Understanding these allows us to analyze the trajectory and determine when the object reaches the ground.

Solving for Time of Flight

The time of flight is found by determining when the object reaches the ground level (y = 0). Setting the vertical position equation to zero and solving for t gives the duration the packet is in the air. This time is crucial to calculate the horizontal distance traveled before impact.

Inverse Problem in Projectile Motion

An inverse problem involves finding the initial conditions needed to achieve a desired outcome. Here, we must find the release point (horizontal distance before the target) so the packet lands exactly on the target. This requires working backward from the target position using the time of flight and horizontal velocity.