Motion in a Plane with Constant Acceleration
In this section, we examine the motion of a particle moving in the x-y plane with a constant acceleration. The primary focus is on how the velocity and position of the object change over time due to this steady acceleration.
Key Relationships:
- Velocity: The change in velocity (v) from an initial velocity (v0) after a time interval (t) can be expressed as:
- \[ v = v_0 + a t \]
where \( a \) is the constant acceleration, affecting both x and y components of velocity as:
- \[ v_x = v_{0x} + a_x t \]
-
\[ v_y = v_{0y} + a_y t \]
-
Position: The position vector change over time is given by:
- \[ r = r_0 + v_0 t + \frac{1}{2} a t^2 \]
where \( r_0 \) is the initial position. This leads to component-wise representation of position:
- \[ x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2 \]
- \[ y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2 \]
The analysis demonstrates that motion in a plane can be treated as two independent one-dimensional motions along perpendicular axes. Understanding these relationships is crucial for tackling complex two-dimensional motion problems effectively.