Answer
a) The car's speed just before the beginning of the braking period is v1 = a1 * t1.
b) The car travels a distance of s1 = (1/2) * a1 * t1^2 before the driver begins to brake.
c) The total distance traveled by the car is:
s_total = s1 + s2
= (1/2) * a1 * t1^2 + (a1 * t1) * t2 + (1/2) * a2 * t2^2
Step-by-step explanation
Let's break down the problem into different stages to find the answers:
Stage 1: Acceleration from rest for t1 = 1 second
a) How fast is the car going just before the beginning of the braking period?
The velocity after time t1, denoted as v1, can be calculated using the equation:
v1 = u + a1 * t1
Since the car starts from rest (u = 0), the equation simplifies to:
v1 = a1 * t1
Therefore, the car's speed just before the beginning of the braking period is v1 = a1 * t1.
b) How far does the car go before the driver begins to brake?
The distance covered during acceleration can be calculated using the equation:
s1 = u * t1 + (1/2) * a1 * t1^2
Since the car starts from rest (u = 0), the equation simplifies to:
s1 = (1/2) * a1 * t1^2
Therefore, the car travels a distance of s1 = (1/2) * a1 * t1^2 before the driver begins to brake.
Stage 2: Braking with acceleration a2 for t2 seconds
For the second stage, we'll consider the initial velocity and position at the beginning of the braking period as v2 = v1 and s2 = s1, respectively.
c) Using the answers to parts (a) and (b) as the initial velocity and position for the motion of the car during braking, what total distance does the car travel?
During braking, the distance covered can be calculated using the equation:
s2 = v2 * t2 + (1/2) * a2 * t2^2
Substituting the initial velocity v2 = v1 = a1 * t1 and the distance traveled during acceleration s1 = (1/2) * a1 * t1^2, we get:
s2 = (a1 * t1) * t2 + (1/2) * a2 * t2^2
Therefore, the total distance traveled by the car is:
s_total = s1 + s2
= (1/2) * a1 * t1^2 + (a1 * t1) * t2 + (1/2) * a2 * t2^2