Talk:A-level Physics/Forces and Motion/Kinematics

Specification From the OCR GCE Physics A specification. Tick with ✔ when you feel each part of the specification has been covered. Try not to add much more than what is in this list as it would not appear in the exam anyway. Use links to wikipedia for more depth in a topic.

Define displacement, speed, velocity and acceleration. ✔

Use graphical methods to represent distance travelled, displacement, speed, velocity and acceleration.

Find the distance travelled by calculating the area under the graph.

Use the slope of a displacement-time graph to find velocity and of a distance-time graph to find speed.

Use the slope of a velocity-time graph to find acceleration.

Derive, from the definitions of velocity and acceleration, equations which represent uniformly accelerated motion in a straight line.

Use equations which represent uniformly accelerated motion in a straight line, including falling in a uniform gravitational field without air-resistance.

Interpret displacement-time and speed-time graphs for motion with non-uniform acceleration.

Explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction.

I have removed the parts regarding vector notation and calculus. Both of these are outside of the scope of an A-level physics course (somewhat surprisingly) and leaving them in will likely confuse students. I have moved the text here. Don't worry, you haven't typed in vain, Mechanics 1 in "A-level mathematics" would find your contributions useful. --Krackpipe 10:36, 3 September 2005 (UTC)

The derivation of the above formulae, uses the basic definitions of velocity and acceleration, with the help of calculus.

By definiton,


 * 1) $$\  \vec v = d\vec x / dt $$
 * 2) $$\  \vec a=d\vec v/dt    $$
 * 3) $$\ \vec v d \vec v / d \vec x = \vec a $$

Where $$ \vec x $$ is position vector (from an axis), $$ \vec v $$ is velocity, and $$ \vec a $$ is accleration

Assuming $$ \vec a $$ to be constant, since the above formulae are defined under that condition only,

V-T Relation
$$\ \vec a dt = d\vec v $$

$$ \int_{0}^{t} \vec a\, dt = \int_{v_i}^{v_f} d \vec v $$

$$\ \vec at = \vec v_f - \vec v_i $$ or $$\ \vec v_f = \vec v_i + \vec a t $$

where $$ \vec v_f $$ is final velocity, and $$ \vec v_i $$ is initial velocity.

X - T Relation
$$\ d \vec x / d \vec t = /vec v $$

$$\ d \vec x = \vec v dt $$

$$\ d \vec x = (\vec v_i + \vec a t )dt $$

The last step was got by substituting the result from the previous derivation. Intergrating,

$$ \int_{x_i}^{x_f} dx =  \int_{0}^{t} (\vec v_i + \vec a t ), dt $$

$$ \vec x_f - \vec x_f = \vec v_i t + 1/2 \vec a t $$

V-X Relation
$$\ \vec v d \vec v / d \vec x = \vec a $$

$$\ \vec v d \vec v = \vec a d \vec x $$

Intergrating

$$\ \int_{v_i}^{v_f} \vec v, d \vec v = \int_{x_i}^{x_f} \vec a, d \vec x $$

$$\ 1/2(v_f ^2 - v_i ^2) = \vec a. (x_f - x_i) $$

$$\ v_f ^2 = v_i ^ 2 + 2 \vec a. (x_f - x_i) $$

Here $$ x_f - x_i $$ is nothing but the displacement of the body, and is substituted by the term $$ \vec s $$. So the final formula is,

$$\ v_f ^2 - v_i ^2 = 2 \vec a. \vec s $$

One very important thing that must be noted here is that the terms $$ v_f ^2 $$ and $$ v_i ^2 $$ are  not  the vector quantities, but the magnitude of the vector whole squared. Technically, it is: $$ v_f ^2 = \vec v_f. \vec v_f $$, i.e the dot product. Another thing to note is that the term $$ 2 \vec a. \vec s $$ is again the dot product of the acceleration and the displacement.

and

From the equation for velocity, and the two equations for acceleration, you can derive the equations of motion. These equations can be used to solve problems that seem very complicated at first. The equations are:


 * $$ \vec v_f = \vec v_i + \vec a t$$
 * $$ v_f^2= v_i^2+ 2 \vec a . \vec s$$
 * $$ \vec s = \vec v_i t+{\frac {1}{2}} \vec a t^2$$
 * $$ |s| = \frac {( |v_i|+ |v_f|)}{2} \times t$$

Where $$\vec a$$ is acceleration, $$ \vec s$$ is displacement, which can also be written as $$ \vec x_f - \vec x_i $$, $$ \vec x_i $$ and $$ \vec x_f $$ being the initial and final position vectors of the body respectively, $$t$$ is time, $$\vec v_i$$ is initial velocity and $$ \vec v_f$$ is final velocity. Note that these equations only work when an object is moving in a straight line and has constant acceleration.

Note that in the 3rd equation the dot product of the two vectors <