## Uniform linear acceleration

**Introduction**

This topic is about particles which move in a straight line and accelerate uniformly. Problems can vary enormously, so you have to have your wits about you. Problems can be broken down into three main categories:

- Constant uniform acceleration
- Time-speed graphs
- Problems involving two particles

__Constant uniform acceleration__Remember what the following variables represent:

*t*= the time ;

*a*= the acceleration ;

*u*= the initial speed ;

*v*= the final speed ;

*s*= the displacement from where the particle started. When the acceleration is negative, it is sometimes called a deceleration or retardation. For example, an acceleration of –3 ms-2 is the same as a deceleration (or retardation) of 3 ms-2.

• To answer this question, you will need to use the four key formulae intelligently.

• It is important to know the second of these equations off by heart; the others appear on Page 40 of The Mathematical Tables. Secondly, you may be asked to derive either of the last two equations from the first two. Practise this.

• These four formulae will be useful elsewhere (for example when doing Questions 3 and 4 on projectiles and connected particles).

Remember that the above formulae may be used only while the acceleration is uniform. If a particle speeds up, but then travels at a constant speed, and then slows down, the above formulae cannot be used for the entire journey. In these cases we solve the problem by drawing a time-speed graph, with time as the horizontal axis.

There are four key points to remember about time-speed graphs:

• The area between the graph and the time-axis represents the distance travelled.

• These four formulae will be useful elsewhere (for example when doing Questions 3 and 4 on projectiles and connected particles).

**Time-speed graphs**Remember that the above formulae may be used only while the acceleration is uniform. If a particle speeds up, but then travels at a constant speed, and then slows down, the above formulae cannot be used for the entire journey. In these cases we solve the problem by drawing a time-speed graph, with time as the horizontal axis.

There are four key points to remember about time-speed graphs:

• The area between the graph and the time-axis represents the distance travelled.

• The slope of the graph represents the acceleration.

• If a particle starts from rest, then

• If a particle starts from rest, then

*v = at*[i.e. the final speed will be the product of the acceleration and the time.]• If a particle accelerates from rest for time

*t1*with acceleration*a*and immediately decelerates to rest in time*t2*with deceleration*d*, then*t1:t2 = d:a*For example, if the acceleration is 6 ms-2 and the deceleration is 8 ms-2, then

• If particles

• If

• If particle

Common mistakes made in doing this question are:

• Assuming that the particle starts from rest, even though this is not stated in the question.

• Using the formulae where they do not apply.

• Jumping into the question before giving it enough clear thought.

• Not drawing a clear time-speed graph.

• Letting

*t1:t2 = d:a = 8:6 = 4:3*. It follows that of the time will be spent accelerating and of the time will be spent decelerating.**Problems involving two particles**• If particles

*P*and*Q*set off together and later overtake each other, then overtaking will occur when*Sp*=*Sq*. If, however,*P*was 25 metres behind*Q*at the start, then when overtaking occurs,*Sp = Sq + 25*• If

*P*and*Q*are a distance*l*apart and move towards each other, they will meet when*Sp + Sq = l*

• The greatest gap between particles*P*and*Q*occurs when*vp = vq*(because if their speeds are unequal then the gap is either increasing or decreasing)• If particle

*A*sets out and, two seconds later, particle*B*sets out in pursuit, then let*t =*the time which*A*spends on the road and*t - 2 =*the time which*B*spends on the road. (Students will often put*t*+ 2 instead of*t*- 2.)**Common mistakes**Common mistakes made in doing this question are:

• Assuming that the particle starts from rest, even though this is not stated in the question.

• Using the formulae where they do not apply.

• Jumping into the question before giving it enough clear thought.

• Not drawing a clear time-speed graph.

• Letting

*u*represent the speed at two different moments. For example, if a particle travels from*a*to*b*, a distance of 30 m in 4 seconds and then travels from*b*to*c*, a distance of 54 m in a further 3 seconds - how would you find the acceleration? You let*u*= the speed of the particle at*a*(not anywhere else!). Then you form an equation for the journey [*ab*] and another for the journey [*ac*] . These equations will be (using*s = ut +½at2*) :*30 = u(4) +½a(16)*and*84 = u(7)+½a(49)*. You solve these simultaneous equations to find*a*. NB: You do not form an equation for [*bc*], as the initial speed will not be the same*u*as in the first equation. q1._linear_acceleration.docx | |

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