A car moves with uniform velocity of 12m/s for 6 sec. After this it accelerate at 2.0m/s² for 4 sec. It then travel for 2 more sec with uniform velocity . finally car decelerate to stop in 15 sec calculate. (a) the distance travelled in five seconds. (b) avarage velocity for the journey, assuming journey was in straight line.
(a) 12m/s * 5s = 60m
(b) total time: 6+4+2+15 = 27s
total distance: (12*6) + (12*4 + 1/2 * 2 * 4^2) + (20*2) + (20*15 - 1/2 * 4/3 * 15^2) = 326m
To calculate the distance traveled in five seconds, we need to divide the journey into different segments and calculate the distance traveled in each segment separately.
Segment 1: Uniform Velocity of 12 m/s for 6 seconds
The distance traveled in this segment can be calculated using the formula:
Distance = Velocity * Time
In this case, the velocity is 12 m/s and the time is 6 seconds:
Distance1 = 12 m/s * 6 sec = 72 meters
Segment 2: Acceleration at 2.0 m/s² for 4 seconds
To calculate the distance traveled during acceleration, we need to use the formula for distance covered with constant acceleration:
Distance = Initial Velocity * Time + 1/2 * Acceleration * Time^2
In this case, the initial velocity is the final velocity of the previous segment, which is 12 m/s. The time is 4 seconds, and the acceleration is 2.0 m/s²:
Distance2 = 12 m/s * 4 sec + 1/2 * 2.0 m/s² * (4 sec)^2 = 48 meters + 1/2 * 2.0 m/s² * 16 sec² = 48 meters + 16 meters = 64 meters
Segment 3: Uniform Velocity for 2 seconds
The distance traveled during uniform velocity can be calculated using the same formula as in segment 1:
Distance3 = 12 m/s * 2 sec = 24 meters
Segment 4: Deceleration to stop in 15 seconds
Since the car decelerates until it stops, its final velocity will be zero. We can now calculate the distance traveled during deceleration using the formula for distance covered with constant deceleration:
Distance = Initial Velocity * Time + 1/2 * Deceleration * Time^2
In this case, the initial velocity is the velocity before deceleration, which is still 12 m/s. The time for deceleration is given as 15 seconds. However, we need to find the deceleration rate, which can be calculated using the formula:
Deceleration = (Final Velocity - Initial Velocity) / Time
Since the final velocity is zero, we have:
Deceleration = (0 m/s - 12 m/s) / 15 sec = -12 m/s / 15 sec = -0.8 m/s² (negative because it is decelerating)
Now we can calculate the distance:
Distance4 = 12 m/s * 15 sec + 1/2 * (-0.8 m/s²) * (15 sec)^2 = 180 meters - 1/2 * 0.8 m/s² * 225 sec² = 180 meters - 90 meters = 90 meters
(a) The total distance traveled in five seconds is the sum of the distances covered in the first three segments (6 sec + 4 sec + 2 sec):
Distance5 = Distance1 + Distance2 + Distance3 = 72 meters + 64 meters + 24 meters = 160 meters
(b) To find the average velocity, we divide the total distance traveled by the total time taken for the journey:
Total Distance = Distance1 + Distance2 + Distance3 + Distance4 = 72 meters + 64 meters + 24 meters + 90 meters = 250 meters
Total Time = 6 sec + 4 sec + 2 sec + 15 sec = 27 seconds
Average Velocity = Total Distance / Total Time = 250 meters / 27 sec ≈ 9.26 m/s
Therefore, the answers are:
(a) The distance traveled in five seconds is 160 meters.
(b) The average velocity for the journey, assuming a straight line, is approximately 9.26 m/s.
To solve this problem, let's break it down into steps:
Step 1: Calculate the distance covered during the first 6 seconds.
Since the car moves with a uniform velocity of 12 m/s for 6 seconds, the distance covered during this time can be calculated using the formula:
Distance = Velocity × Time
Distance = 12 m/s × 6 s
Distance = 72 meters
So, the distance covered during the first 6 seconds is 72 meters.
Step 2: Calculate the distance covered during the acceleration phase.
During the acceleration phase, the car accelerates at a rate of 2.0 m/s² for 4 seconds. To calculate the distance covered during this time, we can use the formula for distance covered during uniform acceleration:
Distance = Initial Velocity × Time + (1/2) × Acceleration × Time²
Since the initial velocity is 12 m/s and the acceleration is 2.0 m/s², and the time is 4 seconds, we can substitute these values into the formula:
Distance = 12 m/s × 4 s + (1/2) × 2.0 m/s² × (4 s)²
Distance = 48 m + (1/2) × 2.0 m/s² × 16 s²
Distance = 48 m + 16 m
Distance = 64 meters
So, the distance covered during the acceleration phase is 64 meters.
Step 3: Calculate the distance covered during the final 2 seconds.
During the final 2 seconds, the car moves with a uniform velocity. Since the car decelerates to stop in 15 seconds, and 6 seconds were already covered in step 1, there are 15 - 6 - 4 - 2 = 3 seconds left for deceleration.
Let's calculate the deceleration (negative acceleration) required to stop the car:
Using the formula:
Final Velocity = Initial Velocity + (Acceleration × Time)
0 m/s = 12 m/s + (Deceleration × 3 s)
Solving for Deceleration:
Deceleration × 3 s = -12 m/s
Deceleration = -12 m/s ÷ 3 s
Deceleration = -4 m/s²
It's negative because we are decelerating.
Now, using the equation of motion:
Distance = Initial Velocity × Time + (1/2) × Acceleration × Time²
Distance = 12 m/s × 2 s + (1/2) × (-4 m/s²) × (2 s)²
Distance = 24 m + (1/2) × (-4 m/s²) × 4 s²
Distance = 24 m - 32 m
Distance = -8 meters
Since distance cannot be negative, we consider this as 0 meters (the car stops).
So, the distance covered during the final 2 seconds is 0 meters.
Step 4: Calculate the total distance covered during the journey.
To calculate the total distance covered, we sum up the distances from each step:
Total Distance = Distance during the first 6 seconds + Distance during the acceleration phase + Distance during the final 2 seconds
Total Distance = 72 m + 64 m + 0 m
Total Distance = 136 meters
So, the total distance covered during the journey is 136 meters.
Step 5: Calculate the average velocity for the journey.
Average velocity is defined as the total displacement divided by the total time taken. Since the journey is in a straight line and the car eventually returns to its starting point, the displacement is 0 meters. Therefore, the average velocity for the journey is also 0 m/s.
Here are the answers to the specific questions:
(a) Distance traveled in five seconds: The car has already traveled 72 meters during the first 6 seconds. Hence, the distance traveled in the first 5 seconds will be the same, which is 72 meters.
(b) Average velocity for the journey: The average velocity for the journey is 0 m/s, as calculated in Step 5.
Note: It's important to note that the negative value for distance in the final step is a result of the car decelerating in the opposite direction of its initial motion. However, since distance is a scalar quantity, we ignore the direction when calculating the total distance.