A car starts from rest and travels for 4.5 s with a uniform acceleration of +1.8 m/s2. The driver then applies the brakes, causing a uniform acceleration of -1.8 m/s2. The breaks are applied for 1.80 s.
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Well, that driver must be really indecisive. It's like they couldn't make up their mind whether to accelerate or decelerate! But let's crunch the numbers and see what happens.
First, let's calculate the distance the car traveled during the first phase, when it was accelerating with a uniform acceleration of +1.8 m/s². We can use the formula:
distance = initial velocity * time + 0.5 * acceleration * time²
Since the car starts from rest, its initial velocity is 0 m/s. Plugging in the values, we get:
distance = 0 * 4.5 + 0.5 * 1.8 * (4.5)²
Now, time to do some math. Hang on! I'm getting my "abacus" ready...
Calculating...
Calculating...
Ah, here it is! The distance during the first phase is 18.225 meters.
Now, let's move on to the second phase, when the driver applies the brakes and the car decelerates with a uniform acceleration of -1.8 m/s². Again, we can use the same formula, but this time the time is 1.80 s:
distance = 0 * 1.80 + 0.5 * (-1.8) * (1.80)²
Calculating...
Calculating...
And voila! The distance during the second phase is -2.916 meters.
Now, if we sum up the distances from both phases, we get:
18.225 meters + (-2.916) meters = 15.309 meters
So, after all this indecisiveness and back-and-forth, the car ends up traveling a total distance of 15.309 meters.
To solve this problem, we need to break it down into two parts: the motion of the car during acceleration and the motion of the car during deceleration.
First, let's calculate the distance covered during acceleration.
Step 1: Determine the time and acceleration during acceleration.
Time during acceleration = 4.5 s
Acceleration during acceleration = +1.8 m/s^2
Step 2: Calculate the distance covered during acceleration using the formula:
Distance = (Initial Velocity x Time) + (0.5 x Acceleration x Time^2)
Since the car starts from rest (initial velocity = 0), the formula simplifies to:
Distance = 0 + (0.5 x Acceleration x Time^2)
Plugging in the values:
Distance = 0.5 x 1.8 x (4.5)^2
Distance = 0.5 x 1.8 x 20.25
Distance = 18.225 meters
Next, let's calculate the distance covered during deceleration.
Step 1: Determine the time and acceleration during deceleration.
Time during deceleration = 1.80 s
Acceleration during deceleration = -1.8 m/s^2
Step 2: Calculate the distance covered during deceleration using the same formula:
Distance = (Initial Velocity x Time) + (0.5 x Acceleration x Time^2)
Similar to the acceleration case, the car starts with a velocity of 0, so the formula simplifies to:
Distance = 0 + (0.5 x Acceleration x Time^2)
Plugging in the values:
Distance = 0.5 x (-1.8) x (1.80)^2
Distance = 0.5 x (-1.8) x 3.24
Distance = -2.916 meters
Note: The negative sign indicates that the car is moving in the opposite direction during deceleration.
Finally, we can calculate the total distance covered by adding the distances covered during acceleration and deceleration.
Total distance = Distance during acceleration + Distance during deceleration
= 18.225 meters + (-2.916 meters)
= 15.309 meters
Therefore, the car covers a total distance of 15.309 meters.
To solve this problem, we can break it down into two parts: the initial motion when the car starts from rest and accelerates, and the subsequent motion when the brakes are applied.
Part 1: Initial Motion
In this part, the car starts from rest and accelerates with a uniform acceleration of +1.8 m/s^2 for a time of 4.5 s. We need to find the distance traveled during this period.
To find the distance traveled during uniform acceleration, we can use the equation:
𝑑 = 𝑣0𝑡 + 1/2 𝑎𝑡^2
where 𝑑 is the distance traveled, 𝑣0 is the initial velocity (which is 0 in this case), 𝑎 is the acceleration, and 𝑡 is the time.
Plugging in the values, we get:
𝑑 = (0)(4.5) + 1/2 (1.8)(4.5)^2
= 0 + 1/2 (1.8)(20.25)
= 1/2 (1.8)(20.25)
= 18.225 m
So, the car travels a distance of 18.225 m during the initial acceleration.
Part 2: Braking Motion
In this part, the car applies the brakes and decelerates with a uniform acceleration of -1.8 m/s^2 for a time of 1.80 s. We need to find the distance traveled during this period.
Using the same equation as before, but with the negative acceleration value, we can find the distance:
𝑑 = 𝑣0𝑡 + 1/2 𝑎𝑡^2
Plugging in the values, we get:
𝑑 = (0)(1.80) + 1/2 (-1.8)(1.80)^2
= 0 + 1/2 (-1.8)(3.24)
= 1/2 (-1.8)(3.24)
= -2.916 m
Since distance cannot be negative, we take the absolute value of -2.916:
𝑑 = |-2.916|
= 2.916 m
So, the car travels a distance of 2.916 m during the braking period.
Therefore, the total distance traveled by the car is the sum of the distances during the initial acceleration and the braking, which is:
Total distance = 18.225 m + 2.916 m
= 21.141 m
So, the car travels a total distance of 21.141 m.