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The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.
A worldline having constant four-acceleration is a Minkowski-circle i.e. hyperbola (see hyperbolic motion) The scalar product of a particle's four-velocity and its four-acceleration is always 0. Even at relativistic speeds four-acceleration is related to the four-force : F μ = m A μ , {\displaystyle F^{\mu }=mA^{\mu },} where m is the ...
These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): = =, where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration.
Consequently, the acceleration is the second derivative of position, [7] often written . Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. [9]: 1 Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to ...
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges: = + = + = (+) = + (). Here θ i and θ f are, respectively, the initial and final angular positions, ω i and ω f are, respectively, the initial and final angular velocities, and α ...
An equation for the acceleration can be derived by analyzing forces. Assuming a massless, inextensible string and an ideal massless pulley, the only forces to consider are: tension force (T), and the weight of the two masses (W 1 and W 2). To find an acceleration, consider the forces affecting each individual mass.
These equations are often used for the calculation of various scenarios of the twin paradox or Bell's spaceship paradox, or in relation to space travel using constant acceleration. b) The constant, transverse proper acceleration = by can be seen as a centripetal acceleration, [13] leading to the worldline of a body in uniform rotation [43] [44]