To solve this problem, we need to analyze the motion of both the automobile and the car that is already in motion. We can break this down into two parts: first, determining how far the automobile travels before it overtakes the car, and second, calculating the speed of the car at that moment.
Understanding the Scenario
We have two vehicles: the automobile that starts from rest with a constant acceleration of 2.2 m/s², and the car that is already moving. We need to find out two things: the distance the automobile travels before it overtakes the car and the speed of the car at that instant.
Setting Up the Equations
Let’s denote:
- a = acceleration of the automobile = 2.2 m/s²
- u = initial speed of the automobile = 0 m/s (since it starts from rest)
- s_a = distance traveled by the automobile
- t = time taken to overtake the car
For the automobile, we can use the equation of motion:
s_a = ut + (1/2)at²
Since the initial speed (u) is 0, this simplifies to:
s_a = (1/2)at² = (1/2)(2.2)t² = 1.1t²
Distance Traveled by the Car
Now, let's consider the car. We need to know its initial speed to find out how far it travels in the same time period. Let’s denote the initial speed of the car as v_c (we'll assume it’s a constant speed for simplicity). The distance traveled by the car can be expressed as:
s_c = v_c * t
Finding the Overtaking Point
At the point of overtaking, the distances traveled by both vehicles will be equal:
s_a = s_c
Substituting the equations we have:
1.1t² = v_c * t
We can simplify this by dividing both sides by t (assuming t is not zero):
1.1t = v_c
From this, we can express the time in terms of the car's speed:
t = v_c / 1.1
Calculating the Distance
Now, we can substitute this expression for time back into the equation for distance traveled by the automobile:
s_a = 1.1(v_c / 1.1)²
This simplifies to:
s_a = 1.1(v_c² / 1.21) = (1.1 / 1.21)v_c²
Finding the Speed of the Car
To find the speed of the car at the moment it is overtaken, we can use the earlier derived relationship:
v_c = 1.1t
Substituting the expression for t we found earlier:
v_c = 1.1(v_c / 1.1) = v_c
This indicates that the speed of the car remains constant throughout the motion, as expected for a vehicle moving at a constant speed.
Final Thoughts
In summary, the distance the automobile travels before overtaking the car can be expressed in terms of the car's speed, and the speed of the car remains constant at the moment of overtaking. If you have a specific value for the car's speed, you can substitute it into the equations to find the exact distance and speed at the instant of overtaking.
