---
title: "Pursuit & Interception Interactive Lab | Sandbox Physics"
description: "Where should you aim to catch a moving target? Aim toward a future meeting point."
canonical_url: "https://sandboxphysics.com/labs/pursuit-interception/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/pursuit-interception/"
---

M004 · Where to aim

# Pursuit & Interception

**Where should you aim to catch a moving target?** Aim toward a future meeting point. A direct chase keeps turning as the target moves sideways; leading the motion can shorten the route. Steering limits, stale positions and a turning target can make either strategy miss.

Compare chasing, leading and proportional navigation. Move the target, limit the steering and delay its reported position; then solve a straight-target minimum-time challenge.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Two constant-speed point agents move in a plane. Turning is bounded, target observations can be delayed, and the run ends at a specified center-distance threshold. Robot symbols are schematic. This is a kinematic navigation model with no collisions, forces or vehicle dynamics.

Physics tutorial

## A meeting point moves ahead of the target

**Background** A robot that always turns toward its moving target follows a curved chase. A robot that aims at a future meeting point can take a shorter route.

**Why it matters** Try the same launch with three steering rules, then make the target turn or send old position reports. A short route helps only if the robot can turn into it.

Start with the essentials

- **Focus question**: Which part of a successful intercept comes from prediction, and which from turning ability?
- **One-sentence intuition**: A steady sight-line direction with decreasing distance signals a meeting course; pointing at the target is a different condition.

### Core mathematical model

#### Constant speed with bounded steering

$$
\dot{\mathbf p}=v_p\begin{pmatrix}\cos\psi\\\sin\psi\end{pmatrix},\qquad |\dot\psi|\le\Omega_p
$$

The robot changes direction at fixed speed. Its turning-rate limit is a kinematic constraint; no motor torque or tire force is modeled.

#### Geometry from delayed target reports

$$
\begin{aligned}\mathbf r_o&=\mathbf q(t-\tau)-\mathbf p(t)\\\dot\lambda_o&=\frac{r_{o,x}\dot r_{o,y}-r_{o,y}\dot r_{o,x}}{|\mathbf r_o|^2}\end{aligned}
$$

Target reports can be old; the robot knows its current pose. All three feedback rules use the same reports without correcting their age. The target has a defined pre-launch trajectory.

#### A constant-speed realization of proportional navigation

$$
\begin{aligned}V_c&=-\frac{\mathbf r_o\cdot\dot{\mathbf r}_o}{|\mathbf r_o|}\\a_\perp&=N\max(V_c,0)\dot\lambda_o\\\dot\psi&=\mathrm{clip}_{[-\Omega_p,\Omega_p]}(a_\perp/v_p)\end{aligned}
$$

The scalar sight-line-rate command becomes acceleration normal to the robot velocity. This reduced planar realization omits the sensor, attitude and autopilot dynamics of a real navigation system.

#### The earliest possible straight-target meeting

$$
\begin{aligned}|\mathbf r_0+\mathbf v_tT_*|&=v_pT_*+\rho\\\psi_*&=\arg(\mathbf r_0+\mathbf v_tT_*)\end{aligned}
$$

Choose the first nonnegative solution. Before this time, even a straight path at full speed cannot reach the target capture disk. A freely chosen initial heading attains the bound for a constant-velocity target with fresh initial information.

### Common difficulties

#### Pointing at the target can waste distance

**Typical misconception** The shortest route always points straight at the visible target.

**Better mental model** A sideways-moving target leaves that point. A future meeting point accounts for the distance it will travel.

#### A steering limit can spoil a prediction

**Typical misconception** Knowing the meeting point guarantees capture.

**Better mental model** The robot may start in the wrong direction and need more time to turn. The exact minimum-time plan assumes it can choose its launch heading.

#### Delayed reports describe another instant

**Typical misconception** A perfectly accurate position report is always good enough.

**Better mental model** An old report can be accurate about the past. The robot still needs the target current motion to choose a useful course.

### Run the experiment

1. 01

   #### Watch a direct chase

   Choose Aim at now, then replay.

   **What to observe:** The blue route bends as the gold target moves sideways.

2. 02

   #### Aim ahead

   Choose Aim ahead with the same speed and turn limit.

   **What to observe:** The initial correction points toward a meeting place ahead of the target. The path and capture time change.

3. 03

   #### Solve the shortest-course challenge

   Choose Fastest straight course and open the prediction details.

   **What to observe:** The robot starts with the analytically chosen heading. Its capture time reaches the straight-target bound within numerical error.

4. 04

   #### Turn the target

   Choose Target turns, then change the target turning rate.

   **What to observe:** Proportional navigation keeps adjusting from the sight-line rotation. A straight-target minimum-time guarantee no longer applies.

5. 05

   #### Slow the information

   Choose Old position reports and increase the report delay.

   **What to observe:** The controller steers from older target coordinates while the actual target continues moving.

6. 06

   #### Compare on equal terms

   Open the simulated strategy comparison and keep the same physical settings.

   **What to observe:** The three feedback rows use the same initial heading, speed, turn limit and delayed reports. An uncaptured trial reports its minimum separation.

Physical conclusion

The shortest straight-target plan chooses its launch heading in advance. Feedback guidance must also cope with steering limits and the age of its observations.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)