Constant acceleration (suvat) equationsEdexcel A-Level Maths: Revision notes

For motion in a straight line with constant acceleration, the suvat variables are:

• $s$: displacement
• $u$: initial velocity
• $v$: final velocity
• $a$: acceleration
• $t$: time Each equation links four of the five. List the three known values and the one you want, and choose the equation that does not contain the fifth variable.

On a velocity--time graph a uniformly accelerating particle has a straight line from $(0,u)$ to $(t,v)$.

• Gradient gives acceleration: $a=\frac{v-u}{t}$, so $v=u+at.$
• Area of the trapezium gives displacement: $s=\frac12(u+v)t.$
• Substituting $v=u+at$ into the area: $s=ut+\frac12at^2.$
• Substituting $u=v-at$ into the area: $s=vt-\frac12at^2.$
• Eliminating $t$ from $v=u+at$ and $s=\frac12(u+v)t$: $v^2=u^2+2as.$ Derivation can also use calculus: $a$ constant, $v=\int a\,dt=u+at$, then $s=\int v\,dt=ut+\frac12at^2$.

Write down what you are given, what you want, and the missing variable:

• no $s$: $v=u+at$
• no $v$: $s=ut+\frac12at^2$
• no $t$: $v^2=u^2+2as$
• no $a$: $s=\frac12(u+v)t$
• no $u$: $s=vt-\frac12at^2$ Example: a car brakes from $30$ m s$^{-1}$ to rest in $90$ m. No $t$ is involved, so $0=30^2+2a(90)$ and $a=-5$ m s$^{-2}$.

For an object moving freely under gravity, model it as a particle with air resistance neglected, so the acceleration is constant: $g=9.8$ m s$^{-2}$ downwards. Choose a positive direction and keep to it. If upwards is positive, $a=-9.8$. At the highest point $v=0$. If an object returns to its starting level, $s=0$. Example: thrown up at $14$ m s$^{-1}$: greatest height $\frac{14^2}{2\times9.8}=10$ m, and time of flight from $s=0$: $0=14t-4.9t^2$, so $t=2.86$ s.

Two objects: write each object's displacement in terms of $t$, then set them equal when they meet. Example: a car passes at $25$ m s$^{-1}$ as a police car starts from rest with $a=4$: $25t=2t^2$, so $t=12.5$ s. Several stages: the final velocity of one stage is the initial velocity of the next. If an object travels at a constant maximum speed after accelerating, use distance $=$ speed $\times$ time for that stage. A quadratic in $t$ may give two roots; reject any that do not fit the situation.