A car starts from rest and travels along a straight track such that it accelerates at 8 m/s2for 10 s and then decelerates at 1.27 m/s2. Determine the distance (in m) travelled by the car in + = 25 seconds. Round off only on the final answer expressed in three decimal places.
The Correct Answer and Explanation is
To determine the total distance traveled by the car, we need to break the problem into two parts: the acceleration phase and the deceleration phase.
Given Data:
- The car starts from rest, so the initial velocity ( u = 0 \, \text{m/s} ).
- Acceleration during the first phase ( a_1 = 8 \, \text{m/s}^2 ), and the time for this phase is ( t_1 = 10 \, \text{s} ).
- Deceleration during the second phase ( a_2 = -1.27 \, \text{m/s}^2 ), and the total time is ( t_2 = 25 – 10 = 15 \, \text{s} ).
Step 1: Distance traveled during acceleration phase (first 10 seconds)
For the first 10 seconds, the car accelerates at ( 8 \, \text{m/s}^2 ), starting from rest. We can use the following kinematic equation to calculate the distance traveled:
[
d_1 = u t + \frac{1}{2} a t^2
]
Substitute the values:
[
d_1 = 0 \times 10 + \frac{1}{2} \times 8 \times (10)^2 = 4 \times 100 = 400 \, \text{m}
]
Step 2: Velocity at the end of acceleration phase
At the end of the 10 seconds, the velocity of the car will be:
[
v_1 = u + a_1 t_1 = 0 + 8 \times 10 = 80 \, \text{m/s}
]
Step 3: Distance traveled during deceleration phase (next 15 seconds)
During the deceleration phase, the car slows down at ( 1.27 \, \text{m/s}^2 ) for 15 seconds. The initial velocity at the beginning of this phase is ( v_1 = 80 \, \text{m/s} ). To calculate the distance traveled during deceleration, we use the same kinematic equation:
[
d_2 = v_1 t_2 + \frac{1}{2} a_2 t_2^2
]
Substitute the values:
[
d_2 = 80 \times 15 + \frac{1}{2} \times (-1.27) \times (15)^2
]
[
d_2 = 1200 + \frac{1}{2} \times (-1.27) \times 225
]
[
d_2 = 1200 – 142.875 = 1057.125 \, \text{m}
]
Step 4: Total distance traveled
The total distance traveled is the sum of the distances during both phases:
[
d_{\text{total}} = d_1 + d_2 = 400 + 1057.125 = 1457.125 \, \text{m}
]
Final Answer:
The total distance traveled by the car is 1457.125 meters.