A car starts from rest and travels for 8.0 s with a uniform acceleration of +2.4 m/s2. The driver then applies the brakes, causing a uniform acceleration of -1.0 m/s2. If the brakes are applied for 1.0 s, determine each of the following.
To determine each of the following, we can use the equations of motion.
1. The final velocity after traveling for 8.0 s with a uniform acceleration of +2.4 m/s².
The equation to calculate the final velocity (v) is:
v = u + at
where:
v = final velocity
u = initial velocity (which is 0 m/s for a car starting from rest)
a = acceleration
t = time
Substituting the given values:
u = 0 m/s
a = +2.4 m/s²
t = 8.0 s
v = 0 + 2.4 * 8.0
v = 19.2 m/s
Thus, the final velocity after traveling for 8.0 s with a uniform acceleration of +2.4 m/s² is 19.2 m/s.
2. The distance traveled during the first 8.0 s.
The equation to calculate distance (s) when the initial velocity is 0 m/s is:
s = (u * t) + (0.5 * a * t²)
Substituting the given values:
u = 0 m/s
a = +2.4 m/s²
t = 8.0 s
s = (0 * 8.0) + (0.5 * 2.4 * (8.0)²
s = 0 + 0.5 * 2.4 * 64.0
s = 76.8 m
Thus, the distance traveled during the first 8.0 s is 76.8 m.
3. The final velocity after applying the brakes for 1.0 s with a uniform acceleration of -1.0 m/s².
Using the same equation as before:
v = u + at
Substituting the given values:
u = 19.2 m/s
a = -1.0 m/s²
t = 1.0 s
v = 19.2 - 1.0 * 1.0
v = 19.2 - 1.0
v = 18.2 m/s
Thus, the final velocity after applying the brakes for 1.0 s with a uniform acceleration of -1.0 m/s² is 18.2 m/s.
To answer each of the following questions, we will break down the problem into different steps:
1. Calculate the final velocity after the first 8.0 seconds of acceleration with +2.4 m/s^2:
First, we need to find the acceleration in the first 8.0 seconds using the equation:
v = u + at
Where:
v = final velocity
u = initial velocity (which is 0 since the car starts from rest)
a = acceleration (+2.4 m/s^2)
t = time (8.0 s)
Plugging in the values, we get:
v = 0 + (2.4)(8.0)
v = 19.2 m/s
So, the final velocity after the first 8.0 seconds of acceleration is 19.2 m/s.
2. Calculate the distance traveled during the first 8.0 seconds of acceleration with +2.4 m/s^2:
We can use the equation:
s = ut + (1/2)at^2
Where:
s = distance traveled
u = initial velocity
t = time
a = acceleration
Since the initial velocity is 0, we can simplify the equation to:
s = (1/2)at^2
Plugging in the values, we get:
s = (1/2)(2.4)(8.0)^2
s = 76.8 m
So, the distance traveled during the first 8.0 seconds of acceleration is 76.8 m.
3. Calculate the final velocity after applying the brakes for 1.0 second with an acceleration of -1.0 m/s^2:
Using the same equation as before:
v = u + at
Where:
v = final velocity
u = initial velocity (which is the same as the final velocity from the previous step, which is 19.2 m/s)
a = acceleration (-1.0 m/s^2)
t = time (1.0 s)
Plugging in the values, we get:
v = 19.2 + (-1.0)(1.0)
v = 19.2 - 1.0
v = 18.2 m/s
So, the final velocity after applying the brakes for 1.0 second is 18.2 m/s.
4. Calculate the distance traveled during the braking period of 1.0 second:
Using the same equation as before:
s = ut + (1/2)at^2
Where:
s = distance traveled
u = initial velocity (which is the same as the final velocity from the previous step, which is 19.2 m/s)
t = time (1.0 s)
a = acceleration (-1.0 m/s^2)
Plugging in the values, we get:
s = 19.2(1.0) + (1/2)(-1.0)(1.0)^2
s = 19.2 - 0.5
s = 18.7 m
So, the distance traveled during the braking period of 1.0 second is 18.7 m.