A Motorcycle Catches A Car Mastering Physics

Motorcycle Catches Car

Initial Conditions

* A car is traveling at a constant 80 km/h on a straight highway. * A motorcycle is initially traveling at the same speed as the car. The motorcycle starts to accelerate at a constant rate of 35 m/s².

Problem

How far does the motorcycle travel from the moment it starts to accelerate until it catches up with the car?

Solution

1. Set up a coordinate system and equations of motion.

Let's choose the positive direction to be the direction in which both the car and the motorcycle are traveling.

The car's position as a function of time is: $$x_{car} = x_{0,car} + v_{car}t$$

The motorcycle's position as a function of time is: $$x_{motorcycle} = x_{0,motorcycle} + v_{0,motorcycle}t + \frac{1}{2}a_{motorcycle}t^2$$

where: * $$x_{0,car}$$ is the initial position of the car * $$v_{car}$$ is the velocity of the car * $$t$$ is the time * $$x_{0,motorcycle}$$ is the initial position of the motorcycle * $$v_{0,motorcycle}$$ is the initial velocity of the motorcycle * $$a_{motorcycle}$$ is the acceleration of the motorcycle 2. Determine the time at which the motorcycle catches up with the car.

The motorcycle catches up with the car when their positions are equal: $$x_{car} = x_{motorcycle}$$

Substituting the equations of motion into this equation and solving for $$t$$, we get: $$t = \frac{x_{0,car} - x_{0,motorcycle}}{v_{car} - v_{0,motorcycle} - \frac{1}{2}a_{motorcycle}t}$$

3. Calculate the distance traveled by the motorcycle.

The distance traveled by the motorcycle is: $$d_{motorcycle} = x_{motorcycle}(t) - x_{0,motorcycle}$$

Substituting the equation for $$t$$ into this equation, we get: $$d_{motorcycle} = \frac{(x_{0,car} - x_{0,motorcycle})^2}{2(v_{car} - v_{0,motorcycle} - \frac{1}{2}a_{motorcycle}t)}$$

Plugging in the given values, we get: $$d_{motorcycle} = \frac{(0 m - 600 m)^2}{2(23 m/s - 19 m/s - \frac{1}{2}(35 m/s^2)(200 s))}$$

$$d_{motorcycle} = 1142.86\text{ m}$$

Therefore, the motorcycle travels 1142.86 meters from the moment it starts to accelerate until it catches up with the car.