1.3 Constant Acceleration

When the acceleration of an object is constant, we can derive some useful equations for simple motion. Starting from equation 1.11 above,

	$\displaystyle\Delta v$	$\displaystyle=\int_{t_{1}}^{t_{2}}a\operatorname{\mathrm{d}}\!t$		(1.12)
	$\displaystyle\implies v_{2}-v_{1}$	$\displaystyle=a\cdot(t_{2}-t_{1}),$		(1.13)

now setting $t_{1}=0,t_{2}=t,v_{1}=v(0)=u,v_{2}=v(t)$ , we get

v(t)=u+at,

(1.14)

where $u$ is the initial velocity of the object. Now we substitute equation 1.14 into equation 1.8 to get

	$\displaystyle\Delta x$	$\displaystyle=\int_{t_{1}}^{t_{2}}(u+at)\operatorname{\mathrm{d}}\!t$		(1.15)
	$\displaystyle\implies x_{2}-x_{1}$	$\displaystyle=u\cdot(t_{2}-t_{1})+\frac{1}{2}a\cdot(t_{2}^{2}-t_{1}^{2}).$		(1.16)

Setting $t_{1}=0,t_{2}=t,x_{1}=x(0)=x_{0},x_{2}=x(t)$ like before and recalling $s(t)=x(t)-x_{0}$ , we get

s(t)=ut+\frac{1}{2}at^{2}.

(1.17)

Finally, squaring equation 1.14 and substituting in equation 1.17 gives

v^{2}(t)=u^{2}+2as(t).

(1.18)

Equations 1.14, 1.17, and 1.18 are known as the SUVAT equations, you probably learned them in school. They are the equations of motion for a object under constant acceleration, i.e. all problems involving constant acceleration in a straight line are solved by them.

Example 1.1.

Ball thrown in the air

Now let’s do some examples where we do not have constant acceleration.

Example 1.2.

Car accelerating then decelerating

v(t)=-\frac{1}{2}t^{4}+3t^{3}

(1.19)

Example 1.3.

Block sliding down a hill (no friction)