A car is stopped at a stoplight. When the light turns green, it accelerates with a constant acceleration of 2.5 m/s^2. At the same instant, a truck passes the car in the same direction with a constant velocity of 15 m/s. how long until the car catches the truck?
To solve this problem, we need to find the time it takes for the car to catch up with the truck. We can do this by finding the distance each vehicle travels and using the formula speed = distance/time.
Let's assume the time it takes for the car to catch the truck is 't'.
First, let's find the distance traveled by each vehicle during this time 't':
Distance traveled by the car = Initial velocity of the car × time + 0.5 × acceleration × time^2
The initial velocity of the car is 0 (since it is at rest when the light turns green), and the acceleration is given as 2.5 m/s^2. Therefore, the distance traveled by the car is: d_car = 0.5 × 2.5 × t^2 = 1.25t^2.
Distance traveled by the truck = constant velocity of the truck × time = 15t.
The car catches the truck when the distance traveled by the car is equal to the distance traveled by the truck:
1.25t^2 = 15t.
We can now solve this equation to find 't':
1.25t^2 - 15t = 0.
Factoring out 't' gives us:
t(1.25t - 15) = 0.
From this equation, either t = 0 (which is not possible) or 1.25t - 15 = 0.
Solving 1.25t - 15 = 0 gives us:
1.25t = 15.
Dividing both sides by 1.25 gives us:
t = 12.
Therefore, it will take 12 seconds for the car to catch the truck.