A car starts from rest and travels for 5.4 s with a uniform acceleration of +1.6 m/s 2 . The driver then applies the brakes, causing a uniform acceleration of -2.2 m/s 2 . The breaks are applied for 1.80 s.
(a) How fast is the car going at the end of the braking period?
(b) How far has the car gone from its start?
s1=a1•t²/2 = 1.6•5.4²/2 =23.33 m.
v=a1•t = 1.6•5.4= 8.64 m/s.
v1= v-a2•t = 8.64 – 2.2•1.8 = 4.68 m/s.
s2= v•t –a2•t²/2 = 8.64•1.8 -2.2•1.8²/2 =15.55- 0.324 = 15.226 m.
s=s1+s2 = 15.226 +23.33=38.556 m.
To find the speed of the car at the end of the braking period, we need to find the final velocity after both the acceleration and braking phases.
(a) To find the final velocity after the acceleration phase, we use the formula:
v = u + at
Where:
v = final velocity
u = initial velocity (which is 0 since the car starts from rest)
a = acceleration of the car during the acceleration phase (+1.6 m/s^2)
t = time taken during the acceleration phase (5.4 s)
Substituting the values into the formula:
v = 0 + (1.6 m/s^2) * (5.4 s)
v = 8.64 m/s
Now, to find the final velocity after the braking phase, we use the same formula but with a different acceleration value:
v = u + at
Where:
v = final velocity
u = initial velocity (which is the velocity after the acceleration phase, i.e., 8.64 m/s)
a = acceleration of the car during the braking phase (-2.2 m/s^2)
t = time taken during the braking phase (1.80 s)
Substituting the values into the formula:
v = 8.64 m/s + (-2.2 m/s^2) * (1.80 s)
v = 8.64 m/s - 3.96 m/s
v = 4.68 m/s
Therefore, the car is going at a speed of 4.68 m/s at the end of the braking period.
(b) To find the distance traveled by the car from its start, we need to find the total displacement during the acceleration and braking phases.
The distance traveled during the acceleration phase can be calculated using the equation:
s = ut + (1/2)at^2
Where:
s = distance traveled during the acceleration phase
u = initial velocity (which is 0 since the car starts from rest)
a = acceleration of the car during the acceleration phase (+1.6 m/s^2)
t = time taken during the acceleration phase (5.4 s)
Substituting the values into the formula:
s = 0 + (1/2) * (1.6 m/s^2) * (5.4 s)^2
s = 0 + 0.5 * 1.6 m/s^2 * 29.16 s^2
s = 23.328 m
The distance traveled during the braking phase can also be calculated using the same equation, but with a different acceleration value:
s = ut + (1/2)at^2
Where:
s = distance traveled during the braking phase
u = initial velocity (which is the velocity after the acceleration phase, i.e., 8.64 m/s)
a = acceleration of the car during the braking phase (-2.2 m/s^2)
t = time taken during the braking phase (1.80 s)
Substituting the values into the formula:
s = (8.64 m/s) * (1.80 s) + (1/2) * (-2.2 m/s^2) * (1.80 s)^2
s = 15.552 m - 1.98 m
s = 13.572 m
Therefore, the car has traveled a total distance of 23.328 m + 13.572 m = 36.900 m from its start.