Question

The equation for the motion of an object with constant acceleration is d = d₀ + vt + ½ at² where d is the distance from a given point in meters, d0 is the initial distance from the starting point in meters, v is the starting velocity in meters per second, a is acceleration in meters per second squared, and t is time in seconds. A car is stopped at a traffic light. When the light turns green, the driver begins to drive, accelerating at a constant rate of 4 meters per second squared. A bus is traveling at a speed of 15 meters per second in another lane. The bus is 7 meters behind the car as the car begins to accelerate. Find when the bus passes the car, when the car passes the bus, and how far each has traveled each time they pass one another. Please Explain

Answer 1

Certainly!

The question states that a car, starting from rest, begins to accelerate at a constant rate of 4 meters per second squared when the traffic light turns green. At the same moment, there's a bus traveling at a constant speed of 15 meters per second, 7 meters behind the car.

Let's first find when the bus passes the car. Since the bus is initially behind, it needs to cover the 7m difference and any distance the car has traveled in the meantime. We're looking for the time 't' when the distances covered by both the car and bus are equal. For this, we can set the two distance equations (distance travelled by car = distance traveled by bus) relative to their initial positions, speeds and accelerations, and solve for 't'.

Upon solving for 't', we get two times, let's take the second time which is 7 seconds. The first time (at t = 0) just represents the initial condition when the car starts accelerating from the traffic light. The second time represents the moment when the bus, which was initially behind the car, catches up to the car.

Now, let's find when the car passes or overtakes the bus. This happens when the speed of the car equals and then surpasses that of the bus. So we equate the car's velocity (which is acceleration * time) to the speed of the bus and solve for 't' again. Doing this, we find that the car will overtake the bus after 3.75 seconds.

Next, we'll find out the distances covered by both car and bus when they pass each other.

Firstly, the distance covered by the bus when it passes the car can be found by substituting the time (t = 7 seconds) into the bus distance equation. The result is 98 meters. This means, when the bus passes the car, it has already covered 98 meters.

Secondly, the distance covered by the car when it passes the bus is computed by substituting the time (t = 3.75 seconds) into the car distance equation, which gives us 28.125 meters. This means, when the car overtakes the bus, it has covered this distance.

So to summarize, the bus overtakes the car at 7 seconds covering 98 meters, while the car overtakes the bus at 3.75 seconds covering 28.125 meters.