Task
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
\[\tag{10} T = \frac{ 500m }{ 18s \cdot 2.4 \frac{ m }{ s^2 }} \]
\[\tag{11} T = 11.57 s \]
