To determine the instantaneous acceleration, take the average acceleration
over smaller and smaller increments of time, giving
a = dv/dt
a = dv/dt
Rectangular Coordinate System
Let the origin O reside at the origin of the Cartesian axis. Resolve
the position vector r(t) into rectangular x, y, and z components,
r = xi + yj + zk
Differentiate to find velocity, giving
To find the acceleration, take a derivative of the velocity, which gives,
Rectangular motion is useful if ax depends only on t, x, and/or vx
(likewise for ay and az).
Note that the equations describing the motion in each coordinate direction
are identical to those of linear motion, as denoted in the Position,
Velocity, Acceleration section.
2-D Projectile Problems
Most problems involve projectiles with constant gravity acceleration in only the y direction. In this special case, the rectilinear motion equations become (assumes up is positive y),
y direction (gravity)
x direction (no gravity)
vy(t) = vyo - g (t - to)
vx(t) = vxo
y(t) = yo + vyo (t - to) - g (t - to)2 / 2
x(t) = xo + vxo (t - to)
vy2 = vo2 -
2 g (x - xo)
vx2 = vxo2
Practice Homework and Test problems now available in the 'Eng Dynamics' mobile app
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