Physics with Calculus Study Guide: Kinematics, Forces, and Dynamics

Q2.65. A car and a truck start from rest at the same instant, with the car initially at some distance behind the truck. The truck has a constant acceleration of 2.10 m/s², and the car has an acceleration of 3.40 m/s². The car overtakes the truck after the truck has moved 60.0 m.

Background

Topic: Kinematics with Constant Acceleration

This question tests your ability to apply kinematic equations to solve for time, initial distance, and velocities when two objects accelerate from rest and one overtakes the other.

Key Terms and Formulas

• Displacement: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity: $v = v_0 + a t$

• Both objects start from rest: $v_0 = 0$

Step-by-Step Guidance

1. Write the displacement equation for both the truck and the car, noting their respective accelerations and initial positions.

2. Set the car's displacement equal to the truck's displacement plus the initial distance between them when the car overtakes the truck.

3. Use the given information that the truck has moved 60.0 m when the car overtakes it to set up your equations.

4. Express the time it takes for the car to overtake the truck in terms of the known accelerations and the truck's displacement.

5. Set up the equations to solve for the initial distance between the car and the truck, and the speeds of each vehicle at the moment they are abreast.

Try solving on your own before revealing the answer!

Final Answer:

(a) Time for the car to overtake the truck: $t = 5.33$ s

(b) Initial distance behind: $d = 24.0$ m

(c) Speed of truck: $v_{truck} = 11.2$ m/s; Speed of car: $v_{car} = 18.1$ m/s

(d) The position vs. time graph will show both vehicles starting at different positions, with the car's curve overtaking the truck's after 60 m.