A-level Physics (Advancing Physics)/Kinematics

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Kinematics is the study of how objects move. One needs to understand a situation in which an object changes speed, accelerating or decelerating, and travelling a certain distance. There are four equations you need to be able to use which relate these quantities.

Contents

[edit] Variables

Before we can understand the kinematic equations, we need to understand the variables involved. They are as follows:

  • t is the length of the interval of time being considered, in seconds.
  • v is the speed of the object at the beginning of the time interval, in ms-1.
  • u is the speed of the object at the end of the time interval, in ms-1.
  • a is the acceleration of the object during the time interval, in ms-2.
  • s is the displacement (distance travelled) of the object during the time interval, in metres.

[edit] Equations

The four equations are as follows:

1. v = u + at\,

2. s = \frac{u + v}{2} t

3. s = ut + \frac{at^2}{2}

4. v^2 = u^2 + 2as\,

[edit] Derivations

It is also useful to know where the above equations come from. We know that acceleration is equal to change in speed per. unit time, so:

a = \frac{v-u}{t} (*)

at = v - u\,

v = u + at\, (1)

We also know that the average speed over the time interval is equal to displacement per. unit time, so:

\frac{u + v}{2} = \frac{s}{t}

s = \frac{u + v}{2} t (2)

If we substitute the value of v from equation 1 into equation 2, we get:

s = \frac{u + (u + at)}{2}t = \frac{2u + at}{2}t = t(u + \frac{at}{2}) = ut + \frac{at^2}{2} (3)

If we take the equation for acceleration (*), we can rearrange it to get:

at = v - u\,

t = \frac{v - u}{a}

If we substitute this equation for t into equation 2, we obtain:

s = \frac{u + v}{2}\frac{v - u}{a} = \frac{(v + u)(v - u)}{2a} = \frac{v^2 - u^2}{2a}

2as = v^2 - u^2\,

v^2 = u^2 + 2as\, (4)

[edit] Questions

1. A person accelerates from a speed of 1 ms-1 to 1.7 ms-1 in 25 seconds. How far has he travelled in this time?

2. A car accelerates at a rate of 18 kmh-2 to a speed of 60 kmh-1, travelling 1 km in the process. How fast was the car travelling before it travelled this distance?

3. A goose in flight is travelling at 4 ms-1. It accelerates at a rate of 1.5 ms-2 for 7 seconds. What is its new speed?

4. How far does an aeroplane travel if it accelerates from 400 kmh-1 at a rate of 40 kmh-2 for 1 hour?

Worked Solutions

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