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A car starts moving rectilinearly, first with acceleration ω = 5.0 m/s2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate ω, comes to a stop. The total time of motion equals τ = 25 s. The average velocity during that time is equal to (v) = 72 km per hour. How long does the car move uniformly?
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A car starts moving rectilinearly, first with acceleration ω = 5.0 m/s...
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A car starts moving rectilinearly, first with acceleration ω = 5.0 m/s...
Analysis:
We are given the following information:
- Initial acceleration (ω) = 5.0 m/s^2
- Initial velocity (v0) = 0 m/s
- Total time of motion (τ) = 25 s
- Average velocity (v) = 72 km/h
We need to determine the duration for which the car moves uniformly.
Step 1: Convert average velocity to m/s
Given that the average velocity is 72 km/h, we need to convert it to m/s to match the units used for acceleration and time.
1 km/h = (1000 m / 3600 s) = 0.2778 m/s
Therefore, the average velocity (v) = 72 km/h * 0.2778 m/s = 20 m/s.
Step 2: Determine the distance traveled
Using the formula for average velocity, we can calculate the distance traveled by the car.
Average velocity (v) = total distance / total time
Total distance = average velocity * total time
Total distance = 20 m/s * 25 s = 500 m
Step 3: Determine the time taken for acceleration and deceleration
During the acceleration and deceleration phases, the velocity changes from 0 m/s to some final velocity and then back to 0 m/s. The total distance covered during these phases can be determined using the equations of motion.
For acceleration:
Final velocity (v) = initial velocity (v0) + acceleration (ω) * time
0 m/s = 0 m/s + 5.0 m/s^2 * t1
t1 = 0 s (since the car starts from rest)
For deceleration:
Final velocity (v) = initial velocity (v0) + acceleration (ω) * time
0 m/s = v1 + 5.0 m/s^2 * t2
v1 = -5.0 m/s * t2
The total distance covered during acceleration and deceleration can be calculated using the equation:
Total distance = (initial velocity + final velocity) * time / 2
Total distance = (0 m/s + v1) * t2 / 2 + (0 m/s + 0 m/s) * t1 / 2
Total distance = v1 * t2 / 2
Substituting the value of v1 from the deceleration equation:
Total distance = (-5.0 m/s * t2) * t2 / 2
Total distance = -2.5 m/s * t2^2
Step 4: Determine the time taken for uniform motion
The remaining distance covered by the car is during the uniform motion phase.
Total distance = distance during acceleration + distance during uniform motion + distance during deceleration
500 m = 0 m + distance during uniform motion + (-2.5 m/s * t2^2)
Simplifying the equation:
500 m = -2.5 m/s * t2^2
Key Point:
Since time cannot be negative, the negative sign is ignored for simplification purposes.
Step 5: Solve for time
Rearranging the equation:
t2^2 = 500 m / 2.5 m
We are given the following information:
- Initial acceleration (ω) = 5.0 m/s^2
- Initial velocity (v0) = 0 m/s
- Total time of motion (τ) = 25 s
- Average velocity (v) = 72 km/h
We need to determine the duration for which the car moves uniformly.
Step 1: Convert average velocity to m/s
Given that the average velocity is 72 km/h, we need to convert it to m/s to match the units used for acceleration and time.
1 km/h = (1000 m / 3600 s) = 0.2778 m/s
Therefore, the average velocity (v) = 72 km/h * 0.2778 m/s = 20 m/s.
Step 2: Determine the distance traveled
Using the formula for average velocity, we can calculate the distance traveled by the car.
Average velocity (v) = total distance / total time
Total distance = average velocity * total time
Total distance = 20 m/s * 25 s = 500 m
Step 3: Determine the time taken for acceleration and deceleration
During the acceleration and deceleration phases, the velocity changes from 0 m/s to some final velocity and then back to 0 m/s. The total distance covered during these phases can be determined using the equations of motion.
For acceleration:
Final velocity (v) = initial velocity (v0) + acceleration (ω) * time
0 m/s = 0 m/s + 5.0 m/s^2 * t1
t1 = 0 s (since the car starts from rest)
For deceleration:
Final velocity (v) = initial velocity (v0) + acceleration (ω) * time
0 m/s = v1 + 5.0 m/s^2 * t2
v1 = -5.0 m/s * t2
The total distance covered during acceleration and deceleration can be calculated using the equation:
Total distance = (initial velocity + final velocity) * time / 2
Total distance = (0 m/s + v1) * t2 / 2 + (0 m/s + 0 m/s) * t1 / 2
Total distance = v1 * t2 / 2
Substituting the value of v1 from the deceleration equation:
Total distance = (-5.0 m/s * t2) * t2 / 2
Total distance = -2.5 m/s * t2^2
Step 4: Determine the time taken for uniform motion
The remaining distance covered by the car is during the uniform motion phase.
Total distance = distance during acceleration + distance during uniform motion + distance during deceleration
500 m = 0 m + distance during uniform motion + (-2.5 m/s * t2^2)
Simplifying the equation:
500 m = -2.5 m/s * t2^2
Key Point:
Since time cannot be negative, the negative sign is ignored for simplification purposes.
Step 5: Solve for time
Rearranging the equation:
t2^2 = 500 m / 2.5 m
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A car starts moving rectilinearly, first with acceleration ω = 5.0 m/s2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate ω, comes to a stop. The total time of motion equals τ = 25 s. The average velocity during that time is equal to (v) = 72 km per hour. How long does the car move uniformly?
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