For our tech demo 3, Stone and I decided to make a catapult that would launch an object when a sensor notified the servo to move.  The sensor detected the distance between itself and a moving object and determined when to shoot.  We laser cut a base, 2 walls to hold a metal rod that would allow the catapult to actively and frictionlessly throw the ball, and  the catapult (thrower) itself.  The spring in the front allows for tension to build as a weight with a string attached keeps the catapult from moving.  When the servo moves, however, it pulls the weight from the catapult and the tension takes over and whips the catapult into action.

// ReadSonar - measure distance using a HC-SR04 or compatible ultrasonic ranger
//
// With the help of Garth Zieglin
================================================================================
// Define constant values.
#include <Servo.h>
// The wiring assignment.
const int TRIG_PIN = 8;
const int ECHO_PIN = 7;
const int SERVO_PIN = 9;
Servo servo;

// The rated distance limit of the sensor, in cm.
const int MAX_DISTANCE = 450;

// A typical speed of sound, specified in cm/sec.
const long SOUND_SPEED = 34000;

const long TIMEOUT = (2 * MAX_DISTANCE * 1000000)/SOUND_SPEED;

void setup()
{
// Initialize the serial UART at 9600 bits per second.
Serial.begin(9600);
servo.attach(SERVO_PIN);
servo.write(0);

// Initialize the trigger pin for output.
pinMode(TRIG_PIN, OUTPUT);
digitalWrite(TRIG_PIN, LOW);

// Initialize the echo pin for input.
pinMode(ECHO_PIN, INPUT);
}
// ================================================================================
// Run one iteration of the main event loop. The Arduino system will call this
// function over and over forever.
void loop()
{

long duration = ping_sonar(); // function is defined below

if (duration > 0) {
float distance = (duration * 1e-6 * SOUND_SPEED) / 2;
if(distance < 10){
servo.write(180);
}

} else {
// if no pulse detected
Serial.println("No ping.");
}

// Allow a little extra time for the sonar to recover.
delay(30);
}

long ping_sonar(void)
{
// Generate a short trigger pulse.
digitalWrite(TRIG_PIN, HIGH);
delayMicroseconds(10);
digitalWrite(TRIG_PIN, LOW);

// Measure the pulse length
return pulseIn(ECHO_PIN, HIGH, TIMEOUT);
}

Source:  Course Website (for parts of the code)