To solve this problem, we can break it down into three stages: acceleration, constant velocity, and deceleration. Let's calculate the time taken for each stage and then sum them up to find the total time.
Stage 1: Acceleration
We are given: Acceleration, a = 2.5 m/s² Final velocity, vf = 20 m/s
We can use the equation of motion: vf = vi + at
Where: vi is the initial velocity (which is 0 in this case since the car starts from rest).
Re-arranging the equation, we have: t1 = (vf - vi) / a
Plugging in the values: t1 = (20 - 0) / 2.5 t1 = 8 seconds
Stage 2: Constant Velocity
The car travels at a constant velocity for a certain distance. We know the total distance covered is 500 m. To calculate the time taken for this stage, we can use the equation:
distance = velocity × time
Since the velocity is constant, we can rearrange the equation as follows: time = distance / velocity
Plugging in the values: t2 = 500 m / 20 m/s t2 = 25 seconds
Stage 3: Deceleration
We are given: Acceleration, a = -4 m/s² (negative because it is deceleration) Final velocity, vf = 0 m/s (the car comes to a stop)
Using the same equation of motion: vf = vi + at
Since the final velocity is 0, we have: 0 = vi + (-4)t
Simplifying: vi = 4t
We can substitute this value for vi into the equation for distance: distance = (vi + vf) / 2 * time
Plugging in the values and rearranging: 500 m = (4t + 0) / 2 * t
Simplifying: 500 m = 2t²
Dividing both sides by 2: 250 m = t²
Taking the square root of both sides: t = √250 t ≈ 15.81 seconds
Total Time: The total time is the sum of the times for each stage: total time = t1 + t2 + t3 total time = 8 + 25 + 15.81 total time ≈ 48.81 seconds
Therefore, it will take approximately 48.81 seconds to cover the total distance of 500 m.