Physics of Dart Throwing

1. Defining Variables

Let's define the key variables involved in the physics of a dart throw:

$ m $: Mass of the dart.
$ v $: Initial velocity of the dart when released by the player.
$ C_d $: Drag coefficient of the dart, which depends on its shape.
$ A $: Cross-sectional area of the dart perpendicular to its velocity.
$ \rho $: Density of air.
$ g $: Acceleration due to gravity ($ g \approx 9.8 \, \text{m/s}^2 $).
$ d $: Distance from the player to the dartboard.
$ h $: Height difference between the release point and the center of the target.
$ \text{COM} $: Center of mass of the dart.
$ L $: Length of the dart.
$ \theta $: Launch angle of the dart with respect to the horizontal.

2. Initial Velocity and Launch Angle

The initial velocity $ v $ and angle $ \theta $ at which the dart is thrown determine its horizontal and vertical velocity components:

Horizontal velocity: $ v_x = v \cdot \cos(\theta) $
Vertical velocity: $ v_y = v \cdot \sin(\theta) $

3. Effect of Gravity on Vertical Motion

In the absence of drag, the dart’s vertical position $ y $ as a function of time $ t $ is given by:

$$ y(t) = v_y \cdot t - \frac{1}{2} g \cdot t^2 $$

At the same time, the horizontal position $ x $ of the dart is:

$$ x(t) = v_x \cdot t $$

4. Including Air Drag

The drag force $ F_d $ acting on the dart as it moves through the air is given by:

$$ F_d = \frac{1}{2} C_d \cdot \rho \cdot A \cdot v^2 $$

This drag force opposes the direction of motion, reducing both the horizontal and vertical velocities. Thus, the equations of motion become:

Horizontal: $ m \frac{dv_x}{dt} = -\frac{1}{2} C_d \cdot \rho \cdot A \cdot v \cdot v_x $
Vertical: $ m \frac{dv_y}{dt} = -m \cdot g - \frac{1}{2} C_d \cdot \rho \cdot A \cdot v \cdot v_y $

These are differential equations that can be solved to find the position and velocity of the dart at any time $ t $, taking drag into account.

5. Center of Mass and Stability of the Dart

The center of mass (COM) of the dart affects its stability during flight. A well-designed dart has its COM closer to the front, which helps it align with its velocity vector and maintain a stable trajectory. The torque $ \tau $ due to the drag force $ F_d $ and the distance from the COM $ r $ is:

$$ \tau = r \cdot F_d $$

The torque will cause the dart to rotate, and the moment of inertia $ I $ of the dart resists this rotation:

$$ \alpha = \frac{\tau}{I} $$ where $ \alpha $ is the angular acceleration.

6. Distance to the Dartboard and Required Accuracy

The dartboard has a standard diameter, and the player stands a fixed distance $ d $ from it. For the dart to hit a specific target on the board, it must be aimed precisely, accounting for the horizontal and vertical deviations caused by gravity and drag. The height difference $ h $ between the release point and the target center must also be considered.

If $ t_f $ is the time taken for the dart to reach the dartboard horizontally, then $ t_f $ can be calculated as:

$$ t_f = \frac{d}{v_x} $$

The vertical position $ y(t_f) $ when the dart reaches the board is given by:

$$ y(t_f) = v_y \cdot t_f - \frac{1}{2} g \cdot t_f^2 $$

The dart will hit the intended spot if $ y(t_f) = h $, which requires careful adjustment of the initial angle $ \theta $ and velocity $ v $ to achieve the desired trajectory.

7. Final Relationship Between Variables

To hit the target, we derive the relationship between the various factors:

$$ h = v \cdot \sin(\theta) \cdot \frac{d}{v \cdot \cos(\theta)} - \frac{1}{2} g \cdot \left( \frac{d}{v \cdot \cos(\theta)} \right)^2 $$

Solving for $ \theta $ gives the required angle, accounting for distance $ d $, gravity $ g $, initial velocity $ v $, and height $ h $.

Conclusion

This derivation shows the complex interaction of drag, gravity, initial conditions, and stability factors like the center of mass and torque on a dart's flight. The final result offers a target trajectory based on these factors, enabling a more accurate dart throw.