At the instant the traffic light turns green, an automobile starts with a constant acceleration a of 2.3 m/s2. At the same instant a truck is 11.7 m down the road and traveling with a constant speed of 10.4 m/s. (a) How far beyond the traffic signal will the automobile overtake the truck? (b) How fast will the automobile be traveling at that instant?
To solve this problem, we need to calculate two things: (a) the distance the automobile travels to overtake the truck, and (b) the speed of the automobile at that instant.
(a) To find the distance the automobile travels to overtake the truck, we can use the equation of motion:
\[s = ut + \frac{1}{2}at^2\]
Where:
s = distance
u = initial velocity
a = acceleration
t = time
For the truck, we know the initial velocity (u) is 10.4 m/s, and since it is traveling at a constant speed, there is no acceleration. Therefore, the equation simplifies to:
\[s_{truck} = u_{truck} \cdot t_{truck}\]
Similarly, for the automobile, we know the initial velocity is 0 m/s since it starts from rest. The acceleration (a) is given as 2.3 m/s^2. Let's assume that the time taken by both the automobile and the truck to reach the point of overtaking is the same. Therefore, the equation for the automobile simplifies to:
\[s_{automobile} = \frac{1}{2}a_{automobile} \cdot t_{automobile}^2\]
To find the time taken by both vehicles to reach the point of overtaking, we can equate the distances traveled by both vehicles and solve for time:
\[s_{truck} = s_{automobile}\]
\[u_{truck} \cdot t_{truck} = \frac{1}{2}a_{automobile} \cdot t_{automobile}^2\]
\[10.4 \cdot t_{truck} = \frac{1}{2} \cdot 2.3 \cdot t_{automobile}^2\]
\[10.4 \cdot t_{truck} = 1.15 \cdot t_{automobile}^2\]
Let's solve this equation to find the value of \(t_{automobile}\).
(b) Once we have the value of \(t_{automobile}\), we can use it to find the speed of the automobile at that instant using the equation:
\[v_{automobile} = u_{automobile} + a_{automobile} \cdot t_{automobile}\]
Now that we have the plan for solving the problem, let's proceed with the calculations.