How fast is the car traveling when it overtakes the truck?

Question

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 3.2 m/s2 . At the same instant a truck, traveling with a constant speed of 20 m/s overtakes and passes the car at the same intersection. The car eventually overtakes the truck. Draw a motion diagram for both the car and the truck.

1. Show ALL equations, then substitution of numerical values with units.

2. If you have a string of numbers with no preceding equation, sketch, or explanation, you will receive NO CREDIT even if it leads to the correct answer.

3. Make sure all symbols used are clearly identified on your sketch. BOX each required answer. Show work in a Logical Order – Don’t jump around.

Answer 1

Begin by writing expressions for the distances traveled by the car and the truck. For the car, initial speed is zero and acceleration is 3.2 m/s^2, so d = v i * t + 1/2 a * t^2 yields d (car) = 1.6 t^2.

For the truck, v has a constant value of 20 m/s, do d (truck) = 20 * t.

To say that the car catches the truck is to say that the car travels the same distance as the truck. Mathematically, d (car) = d (truck), so 1.6 t^2 = 20 * t. Solving for t gives t = 12.5 s.

Now we can calculate the speed of the car at t = 12.5 s. Using v = vi + at, with vi = 0, a = 3.2 m/s^2, and t = 12.5 s, we find v = 0 + (3.2 m/s^2)(12.5 s) = 40 m/s.

We can also find where the car catches the truck by substituting t = 12.5 s into either of the distance expressions above. Better yet, substitute this value into both equations to make sure they do in fact travel the same distance.

Below is a graph showing the positions of the car and truck from the start until just after the car passes the truck.