At the instant the trafic light turns green, an automobile starts moving with a constant acceleration of 5.0 m/s2. At the same instant a truck, travelling with a constant speed of 10 m/s, overtakes and passes the automobile. (a) How far beyond the trafic signal will the automobile overtake the truck? (b) How fast will the automobile be at that instant?
To solve this problem, we can use the kinematic equation:
\[v^2 = u^2 + 2as\]
Where:
- v is the final velocity of the automobile
- u is the initial velocity of the automobile
- a is the acceleration of the automobile
- s is the distance covered by the automobile
(a) To find the distance beyond the traffic signal where the automobile overtakes the truck, we need to find the distance covered by the truck at the instant the automobile catches up with it.
The truck is traveling at a constant speed, so its acceleration is zero.
Using the formula \(v^2 = u^2 + 2as\), we can calculate the distance covered by the truck:
\[s_{truck} = \frac{v_{truck}^2 - u_{truck}^2}{2 \cdot a_{truck}}\]
Since the acceleration of the truck is zero, \(a_{truck} = 0\). Therefore, the equation simplifies to:
\[s_{truck} = \frac{v_{truck}^2}{2 \cdot a_{truck}}\]
Substituting the values, we get:
\[s_{truck} = \frac{(10 \, \text{m/s})^2}{2 \cdot 0} = \text{undefined}\]
This means that the truck will never cover any distance since its acceleration is zero. Therefore, the automobile will not overtake the truck.
(b) Since the automobile will not overtake the truck, there is no need to calculate the speed of the automobile at that instant. The automobile will remain behind the truck.