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# Uniformly accelerated movement

A car accelerates with a constant acceleration of a = 2,4 m/s². How long does it take for the car from 0 to 100 km/h?

### Solution

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The given acceleration is the second derivation of x(t):

$\tag{1} \ddot x(t) = a$

Integrated once this yields the speed:

$\tag{2} \int \ddot x(t) dt \xrightarrow{} \dot x(t)$

$\tag{3} \dot x(t) = at+c_1$

The start condition for the moveme

The boundary condition applies that the speed at time t = 0 is zero. The integration constant c1 can thus be eliminated.

$\tag{4} \dot x(t=0) = 0$

$\tag{5} c_1 = 0$

Since the acceleration is given in m/s², the speed is also converted into m/s.

$\tag{6} 100 km/h = \frac{ 100000m }{ 3600 s}$

$\tag{7} 100 km/h = \frac{ 500 m }{ 18 s}$

We are looking for the value t for which the function gives the end speed of 100 km/h:

$\tag{8} \dot x(t=T) = \frac{ 500m }{ 18 s}$

$\tag{9} \frac{ 500m }{ 18s } = a T$

with 2,4 m/s² for the accelerations it yields

$\tag{10} T = \frac{ 500m }{ 18s \cdot 2.4 \frac{ m }{ s^2 }}$

$\tag{11} T = 11.57 s$