Question

a mercedes car can accelerate up to 200 km/h from rest in 10.9 seconds. how far has it gone in this time and what is its acceleration (in m/s²)?

Explanation:

Step1: Convert the final velocity to m/s

We know that \(1\ km = 1000\ m\) and \(1\ h=3600\ s\).
For \(v = 200\ km/h\), \(v=\frac{200\times1000}{3600}\ m/s=\frac{2000}{36}\ m/s=\frac{500}{9}\ m/s\approx55.56\ m/s\)

Step2: Calculate the acceleration

Use the formula \(v = u+at\). Since \(u = 0\) (starts from rest), \(a=\frac{v - u}{t}\)
Substitute \(v=\frac{500}{9}\ m/s\), \(u = 0\) and \(t = 10.9\ s\)
\(a=\frac{\frac{500}{9}-0}{10.9}=\frac{500}{9\times10.9}=\frac{500}{98.1}\approx5.1\ m/s^{2}\)

Step3: Calculate the distance traveled

Use the formula \(s=ut+\frac{1}{2}at^{2}\). Since \(u = 0\), \(s=\frac{1}{2}at^{2}\)
Substitute \(a=\frac{500}{98.1}\ m/s^{2}\) and \(t = 10.9\ s\)
\(s=\frac{1}{2}\times\frac{500}{98.1}\times(10.9)^{2}\)
First, \((10.9)^{2}=118.81\)
\(s=\frac{1}{2}\times\frac{500}{98.1}\times118.81=\frac{500\times118.81}{2\times98.1}\)
\(s=\frac{59405}{196.2}\approx303\ m\)

Answer:

The distance traveled is approximately \(303\ m\) and the acceleration is approximately \(5.1\ m/s^{2}\)