At the instant a traffic light turns green, a car starts with a constant acceleration of 1.8 m/s(squared). At the same instant, a truck travelling at a constant velocity of 8.5 m/s passes the car. How far beyond the starting point will the car catch up to the truck?
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To find the distance beyond the starting point where the car catches up to the truck, we need to determine the time it takes for the car to catch up.
Let's break down the problem into smaller steps:
Step 1: Find the time it takes for the car to catch up to the truck.
Since the truck is traveling at a constant velocity, its position doesn't change over time. Therefore, the car must catch up to the truck by reducing the initial distance between them.
We can use the equation of motion:
\[ s = ut + \frac{1}{2}at^2 \]
Where:
- s is the distance traveled
- u is the initial velocity
- a is the constant acceleration
- t is the time taken
For the truck, the distance traveled is 0, as it is stationary (constant velocity of 8.5 m/s).
For the car, the distance traveled is unknown, and the initial velocity is 0 (as it starts from rest).
Let's denote the time it takes for the car to catch up to the truck as t_catch.
For the truck:
\[ s_{truck} = 0 \]
\[ u_{truck} = 8.5 \, \text{m/s} \]
\[ a = 0 \, \text{m/s}^2 \]
\[ t_catch = ? \]
For the car:
\[ s_{car} = ? \]
\[ u_{car} = 0 \, \text{m/s} \]
\[ a = 1.8 \, \text{m/s}^2 \]
\[ t_catch = ? \]
Since the car and the truck will meet at the same position at the time they meet, we can equate their positions:
\[ s_{car} = s_{truck} \]
Using the equation of motion for the car:
\[ s_{car} = 0 \times t_catch + \frac{1}{2} \times 1.8 \times t_catch^2 \]
\[ s_{car} = 0.9 \times t_catch^2 \]
Substituting this into the equation above, we get:
\[ 0.9 \times t_catch^2 = 0 \]
\[ t_catch^2 = 0 \]
\[ t_catch = 0 \]
From the equation, we can see that the time it takes for the car to catch up is 0 seconds. This tells us that the car catches up to the truck immediately at the instant the traffic light turns green.
Step 2: Find the distance traveled by the car during this time (0 seconds).
Since the car starts from rest and catches up to the truck immediately, it doesn't travel any distance beyond the starting point. Therefore, the answer is 0 meters.
In conclusion, the car catches up to the truck at the same instant the traffic light turns green, and it doesn't travel any distance beyond its starting point.