A race car completes the final lap of a race on a 2-kilometer circular track. At the start of the lap, it crosses the line at 60 meters/second. It completes the lap 28.57 seconds later, crossing the line at 85 meters/second. Calculate the displacement and average acceleration of the car within the span of 12 seconds.
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To calculate the displacement and average acceleration of the race car within the span of 12 seconds, we need to use the given information about its initial and final velocities.
Step 1: Calculate the acceleration of the car.
Acceleration is the change in velocity divided by the time taken. Since we have the initial and final velocities, we can use the formula:
Acceleration = (Final Velocity - Initial Velocity) / Time
Given:
Initial Velocity = 60 meters/second
Final Velocity = 85 meters/second
Acceleration = (85 m/s - 60 m/s) / 28.57 s
Step 2: Calculate the displacement of the car within 12 seconds.
Assuming the acceleration is constant, we can use the equation:
Displacement = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Given:
Initial Velocity = 60 meters/second
Acceleration = (85 m/s - 60 m/s) / 28.57 s
Time = 12 seconds
Displacement = (60 m/s) * (12 s) + (1/2) * [(85 m/s - 60 m/s) / 28.57 s] * (12 s)^2
Now, let's solve the equations to find the answers.
Acceleration = (85 - 60) / 28.57 = 0.87 m/s^2 (rounded to two decimal places)
Displacement = (60 * 12) + (1/2) * (0.87) * (12^2) = 720 + 62.64 = 782.64 meters (rounded to two decimal places)
Therefore, within the span of 12 seconds, the race car's average acceleration is approximately 0.87 m/s^2, and its displacement is approximately 782.64 meters.