Your are sitting in your car, which is standing at the side of the road with the engine idling. In the rearview mirror, you see one of your class mates approaching on a bicycle on the sidewalk. The moment the other student passes you, you start accelerating at 2.67 m/s2, while the bicycle continues to move along at a constant speed of 10.9 m/s. After what distance will you catch up with your classmate?

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To find out after what distance you will catch up with your classmate, we can use the concept of relative motion.

First, let's determine the initial separation between you and your classmate. Since your car is standing still and your classmate is approaching you from behind on a bicycle, the initial separation is equivalent to the distance between your car and your classmate when he passes you.

Next, we need to calculate the time it takes for you to catch up with your classmate. To do this, we can use the equation of motion:

d = ut + (1/2)at^2

Where:
- d is the distance covered
- u is the initial velocity
- a is the acceleration
- t is the time

For your classmate on the bicycle, since he is moving at a constant speed, the acceleration (a) is zero. Therefore, the equation simplifies to:

d = ut

Now we can calculate the time it takes for you to catch up, t. We'll assume that your classmate passed you at time t = 0.

Using the equation for your classmate:

d_classmate = u_classmate * t

For you in the car:

d_you = (1/2)at^2

Since the d_you is the distance traveled when you catch up to your classmate, it is equal to the d_classmate. Therefore:

(1/2)at^2 = u_classmate * t

Rearranging the equation:

(1/2)at^2 - u_classmate * t = 0

Now we can solve this quadratic equation for t. By plugging in the values for acceleration (a) and the classmate's initial velocity (u_classmate), we can find the time it takes for you to catch up with your classmate.

Once we have the time, we can find the distance you traveled (d_you = (1/2)at^2) to catch up to your classmate.