CES Mechanics II Lecture 3
Topics:
- Acceleration, velocity, displacement (cntd) 1.1.3
- Case 3: Acceleration is a function of velocity
This case arises when the forces depend on velocity, for example, in case of hydrodynamic or aerodynamic drag forces. When a = a(v), a formula for position as a function of velocity can be obtained by separation of variables:
x(v) = x_0 + \int_{v_0}^v \frac{\bar{v} d\bar{v}}{a(\bar{v})},
or, using indefinite integral notation:
x(v) = \int \frac{v dv}{a(v)} + C_5,
where C_5 is an arbitrary constant of integration.
The expression for x(v) can be often inverted to obtain v(x).
The next formula can be used to derive time as a function of velocity:
t(v) = t_0 + \int_{v_0}^v \frac{d\bar{v}}{a(\bar{v})},
or, using indefinite integral notation:
t(v) = \int \frac{dv}{a(v)} + C_6,
where C_6 is an arbitrary constant of integration.
The expression for t(v) can be often inverted to obtain v(t).
These formulas are summarized in Table 1.1 in the TM3 book, or in this PDF table.
- Case 3: Acceleration is a function of velocity
- Superposition of rectilinear motion 1.2.2
In general, accelerations in 2D (or 3D) motion will depend in a complex way on all components of velocity and displacement:
\mathbf{a} = \mathbf{a}(\mathbf{v}, \mathbf{r}, t),
or in scalar notation:
a_x = a_x(v_x, v_y, x, y, t),
a_y = a_y(v_x, v_y, x, y, t).
In this case, ODE knowledge is required to find velocity and position of the particle. However, in some cases the motion in different directions takes place independently of one another:
a_x = a_x(v_x, x, t),
a_y = a_y(v_y, y, t).
In such a case, we have a superposition of two (or three) rectilinear motions, which can be independently analyzed using the methods discussed in the previous lecture.
Corresponding Gross et al. Kinetik chapters are shown in blue.
Key techniques:
- deriving velocity and displacement from acceleration
- superposing independent rectilinear motions in several directions
ConcepTests (RWTH only):
