Question

At the instant a traffic light turns green, a car starts from rest with a constant acceleration of 3.0 m/s2. At the same instant, a truck, traveling with a constant speed of 12 m/s, passes the car.

A) How much time does it take for the car to catch up to the truck?
B) How far have the car and truck traveled when the car has caught up?
C) How fast is the car traveling at the point where it catches up to the truck?

Answer 1

A car with a constant acceleration of 3.0 m/s^2 catches up to a truck traveling at a constant speed of 12 m/s. It takes 4 seconds for the car to catch up to the truck, and both the car and truck travel a distance of 24 meters. The car is traveling at a speed of 12 m/s when it catches up to the truck.

Step-by-step explanation:

A) Time taken for the car to catch up to the truck

To find the time it takes for the car to catch up to the truck, we can use the equation:

t = (vf - vi) / a

where t is the time, vf is the final velocity of the car, vi is the initial velocity of the car, and a is the acceleration. Since the car starts from rest, its initial velocity is 0 m/s. The final velocity can be found using the equation:

vf = vi + at

where vi is the initial velocity of the truck, a is the acceleration of the car, and t is the time.

Substituting the given values, we have:

vf = 12 m/s + (3.0 m/s^2)(t)

Setting vf equal to 0 m/s (since the car catches up to the truck), we can solve for t:

0 = 12 m/s + (3.0 m/s^2)(t)

-12 m/s = 3.0 m/s^2(t)

t = -4 s

Since time cannot be negative, we ignore the negative solution. Therefore, it takes the car 4 seconds to catch up to the truck.

B) Distance traveled by the car and truck when the car catches up

To find the distance traveled by the car and truck when the car catches up, we can use the equation:

d = vi*t + 0.5*a*t^2

where d is the distance, vi is the initial velocity, a is the acceleration, and t is the time.

Substituting the given values, we have:

d = (0 m/s)(4 s) + 0.5(3.0 m/s^2)(4 s)^2

d = 0 + 0.5(3.0 m/s^2)(16 s^2)

d = 0 + 24 m

d = 24 m

Therefore, both the car and the truck travel a distance of 24 meters when the car catches up.

C) Speed of the car when it catches up to the truck

The speed of the car when it catches up to the truck is equal to the final velocity of the car, which we can find using the equation:

vf = vi + at

Substituting the given values, we have:

vf = 0 m/s + (3.0 m/s^2)(4 s)

vf = 0 + 12 m/s

vf = 12 m/s

Therefore, the car is traveling at a speed of 12 m/s when it catches up to the truck.