A car starts from rest and travels for t1 seconds with a uniform acceleration a1. The driver then applies the brakes, causing a uniform acceleration a2. If the brakes are applied for t2 seconds, determine the following. Answers are in terms of the variables a1, a2, t1, and t2.
(a) How fast is the car going just before the beginning of the braking period?
(b) How far does the car go before the driver begins to brake?
(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?
To find the answers to these questions, we need to use the equations of motion for uniformly accelerated motion. The three equations we will use are:
1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as
Where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- t is the time taken
- s is the displacement
Let's solve each part of the question:
(a) To find the velocity just before the beginning of the braking period, we need to find the final velocity of the car during the first part of motion.
Using equation 1, we have:
v1 = 0 + a1 * t1
v1 = a1 * t1
So, the car's velocity just before the beginning of the braking period is v1 = a1 * t1.
(b) To find the distance the car goes before the driver begins to brake, we need to find the displacement during the first part of motion.
Using equation 2, we have:
s1 = (0 * t1) + (1/2) * a1 * t1^2
s1 = (1/2) * a1 * t1^2
So, the car goes a distance of s1 = (1/2) * a1 * t1^2 before the driver begins to brake.
(c) To find the total distance the car travels, we need to find the displacement during the braking period.
Using equation 3, we have:
v2^2 = v1^2 + 2 * a2 * s2
(0)^2 = (a1 * t1)^2 + 2 * a2 * s2
0 = a1^2 * t1^2 + 2 * a2 * s2
Rearranging the equation, we get:
s2 = -(a1^2 * t1^2) / (2 * a2)
The total distance the car travels is:
s_total = s1 + s2
s_total = (1/2) * a1 * t1^2 - (a1^2 * t1^2) / (2 * a2)
So, the total distance the car travels is s_total = (1/2) * a1 * t1^2 - (a1^2 * t1^2) / (2 * a2).
To solve this problem, we can use the equations of motion. Let's break it down step by step.
Step 1: Determine the final velocity before braking (v1).
Using the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate v1.
Given:
Initial velocity u = 0 (the car starts from rest)
Acceleration a1 (during the first phase)
Time t1 (duration of the first phase)
Therefore, v1 = u + a1 * t1
v1 = 0 + a1 * t1
v1 = a1 * t1 (Answer to part (a))
Step 2: Determine the distance traveled in the first phase (d1).
Using the equation of motion: d = ut + (1/2)at^2, where d is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time, we can calculate d1.
Given:
Initial velocity u = 0 (the car starts from rest)
Acceleration a1 (during the first phase)
Time t1 (duration of the first phase)
Therefore, d1 = u * t1 + (1/2) * a1 * t1^2
d1 = 0 + (1/2) * a1 * t1^2
d1 = (1/2) * a1 * t1^2 (Answer to part (b))
Step 3: Determine the final velocity after braking (v2).
Using the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate v2.
Given:
Initial velocity u = v1 (the final velocity before braking)
Acceleration a2 (during the second phase)
Time t2 (duration of the second phase)
Therefore, v2 = u + a2 * t2
v2 = v1 + a2 * t2 (Answer to part (a))
Step 4: Determine the distance traveled in the second phase (d2).
Using the equation of motion: d = ut + (1/2)at^2, where d is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time, we can calculate d2.
Given:
Initial velocity u = v1 (the final velocity before braking)
Acceleration a2 (during the second phase)
Time t2 (duration of the second phase)
Therefore, d2 = u * t2 + (1/2) * a2 * t2^2
d2 = v1 * t2 + (1/2) * a2 * t2^2 (Answer to part (b))
Step 5: Determine the total distance traveled (d_total).
To calculate the total distance traveled, we add the distance traveled during the first phase (d1) and the distance traveled during the second phase (d2).
Therefore, d_total = d1 + d2
d_total = (1/2) * a1 * t1^2 + v1 * t2 + (1/2) * a2 * t2^2 (Answer to part (c))
To summarize:
(a) The speed of the car just before the beginning of the braking period is v1 = a1 * t1.
(b) The distance the car goes before the driver begins to brake is d1 = (1/2) * a1 * t1^2.
(c) The total distance the car travels is d_total = (1/2) * a1 * t1^2 + v1 * t2 + (1/2) * a2 * t2^2.
See the 3rd link below under Related Questions.
A similar question was answered for (a).