Equation of motion

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In elementary physics and linear kinematics, the equations of motion are five equations that apply to bodies moving linearly (that is, one dimension) with uniform acceleration. (In advanced physics, the Euler-Lagrange equations, differential equations derived from the Lagrangian, are also called "equations of motion" . The following article is about elementary physics only.)

Contents 1 Linear equations of motion 2 Classic version 2.1 Examples 2.2 Extension 3 Rotational equations of motion 4 Derivation 4.1 Motion equation 1 4.2 Motion equation 2 4.3 Motion equation 3 4.4 Motion equation 4 4.5 Motion equation 5 5 See also 6 References

Linear equations of motion

The body is considered at two instants in time: one "initial" point and one "current". Often, problems in kinematics deal with more than two instants, and several applications of the equations are required.

<math>v_f = v_i + a\Delta t \,</math> <math>d = \begin{matrix} \frac{1}{2} \end{matrix} (v_i + v_f)\Delta t</math> <math>d = v_i\Delta t + \begin{matrix} \frac{1}{2} \end{matrix} a\Delta t^2</math> <math>v_f^2 = v_i^2 + 2ad \,</math> <math>d = v_f\Delta t - \begin{matrix} \frac{1}{2} \end{matrix} a\Delta t^2</math> where... <math>v_i \,</math> is the body's initial speed

and its current state is described by: <math>d \,</math>, the distance travelled from initial state <math>v_f \,</math>, the current speed <math>\Delta t \,</math>, the time between the initial and current states <math>a</math> is the constant acceleration, or in the case of bodies moving under the influence of gravity, g.

Note that each of the equations contains four of the five variables. When using the above formulae, it is sufficient to know three out of the five variables to calculate remaining two.

Classic version

The above equations are often found in the following version:

<math>v = u+at \,</math>

<math>s = \frac {1} {2}(u+v) \cdot t </math>

<math>s = ut + \frac {1} {2} a t^2 </math>

<math>v^2 = u^2 + 2 a s \,</math>

<math>s = vt - \frac {1} {2} a t^2 </math>

where

s = the distance travelled from the initial state to the final state (displacement)

u = the initial speed

v = the final speed

a = the constant acceleration

t = the time taken to move from the initial state to the final state

Examples

Many examples in kinematics involve projectiles, for example a ball thrown upwards into the air.

Given initial speed u, one can calculate how high the ball will travel before it begins to fall.

The acceleration is normal gravity g. At this point one must remember that while these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as uni-directional vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using the 4th equation, we have:

<math>s= \frac{v^2 - u^2}{-2g}</math>

Substituting and cancelling minus signs gives:

<math>s = \frac{u^2}{2g}</math>

Extension

More complex versions of these equations can include a quantity <math>\Delta</math>s for the variation on displacement (s - s₀), s₀ for the initial position of the body, and v₀ for u for consistency.

<math>v = v_0 + at \,</math>

<math>s = s_0 + \begin{matrix} \frac{1}{2} \end{matrix} (v_0 + v)t \,</math>

<math>s = s_0 + v_0 t + \begin{matrix} \frac{1}{2} \end{matrix}{at^2} \,</math>

<math>(v)^2 = (v_0)^2 + 2a \Delta s \,</math>

<math>s = s_0 + v t - \begin{matrix} \frac{1}{2} \end{matrix}{at^2} \,</math>

However a suitable choice of origin for the one-dimensional axis on which the body moves makes these more complex versions unnecessary.

Rotational equations of motion

The analogues of the above equations can be written for rotation:

<math> \omega = \omega_0 + \alpha t \,</math>

<math> \phi = \phi_0

+ \begin{matrix} \frac{1}{2} \end{matrix}(\omega_0 + \omega)t </math>

<math> \phi = \phi_0 + \omega_0 t + \begin{matrix} \frac{1}{2} \end{matrix}\alpha {t^2} \,</math>

<math> (\omega)^2 = (\omega_0)^2 + 2\alpha \Delta \phi \,</math>

<math> \phi = \phi_0 + \omega t - \begin{matrix} \frac{1}{2} \end{matrix}\alpha {t^2} \,</math>

where:

<math>\alpha</math> is the angular acceleration

<math>\omega</math> is the angular velocity

<math>\phi</math> is the angular displacement

<math>\omega_0</math> is the initial angular velocity

<math>\phi_0</math> is the initial angular displacement

<math>\Delta \phi</math> is the variation on angular displacement (<math>\phi</math> - <math>\phi_0</math>).

Derivation

Motion equation 1

By definition of acceleration,

<math>\ a = \frac{v - u}{t}</math>

Hence

<math>at = v - u \,</math>

<math>v = u + at \,</math>

Motion equation 2

By definition,

<math> \mathrm{ average\ velocity } = \frac{s}{t}</math>

Hence

<math> \begin{matrix} \frac{1}{2} \end{matrix} (u + v) = \frac{s}{t}</math>

<math>s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v)t</math>

Motion equation 3

Insert Motion Equation 1 into Motion Equation 2

<math>s = \begin{matrix} \frac{1}{2} \end{matrix} (u + u + at)t</math>

<math>s = \begin{matrix} \frac{1}{2} \end{matrix} (2u + at)t</math>

<math>s = ut + \begin{matrix} \frac{1}{2} \end{matrix} at^2</math>

Motion equation 4

<math>t = \frac{v - u}{a}</math>

Using Motion Equation 2, replace t with above

<math>s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v) ( \frac{v - u}{a} )</math>

<math>2as = (u + v)(v - u) \,</math>

<math>2as = v^2 - u^2 \,</math>

<math>v^2 = u^2 + 2as \,</math>

Motion equation 5

Using Motion Equation 1 to replace u in motion equation 3 gives

<math>s = vt - \begin{matrix} \frac{1}{2} \end{matrix} at^2</math>

See also

• Scalar
• Vector
• Distance
• Displacement
• Speed
• Velocity
• Acceleration
• Jerk
• Angular displacement
• Angular speed
• Angular velocity
• Angular acceleration
• Parabolic trajectory
• Newton's laws of motion
• Torricelli's Equation

References

• Fundamentals of Physics Robert Resnick, David Halliday, Jearl Walker

de:Bewegungsgleichung pl:Kinematyczne równanie ruchu pt:Equações de movimento