At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.80 m/s2. At the same instant a truck, traveling with a constant speed of 20.0 m/s, overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck?

Verified step by step guidance

1. Identify the given values: acceleration of the car, a_c = 2.80 m/s^2, and the constant speed of the truck, v_t = 20.0 m/s.
2. Set up the equations of motion for both the car and the truck. For the car, use the equation s_c = 0.5 * a_c * t^2, where s_c is the distance traveled by the car and t is the time. For the truck, use s_t = v_t * t, where s_t is the distance traveled by the truck.
3. Set the distances equal to each other to find the time when the car overtakes the truck. Solve 0.5 * a_c * t^2 = v_t * t for t.
4. Substitute the value of t back into either the car's or the truck's equation of motion to find the distance at which the car overtakes the truck.
5. Check the units and make sure they are consistent throughout the calculations to ensure the accuracy of the result.

Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In this problem, the car accelerates at 2.80 m/s², meaning its speed increases by 2.80 m/s every second. This concept is crucial for determining the car's position as a function of time using the kinematic equations.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For the car, the equation s = ut + 0.5at² can be used to find the distance it travels, while the truck's distance can be calculated using s = vt, where v is its constant speed.

Relative Motion

Relative motion involves analyzing the movement of one object in relation to another. In this scenario, the car's motion must be compared to the truck's constant speed to determine when they are at the same position. Understanding relative motion is essential for solving problems where two objects start from different conditions and move towards each other.