A car starts from rest and travels for 7.0 s with a uniform acceleration of +3.0 m/s2. The driver then applies the brakes, causing a uniform acceleration of −2.5 m/s2. If the brakes are applied for 2.0 s, determine each of the following.
How far has the car gone?
"A car starts from rest and travels for 7.0 s with a uniform acceleration of +3.0 m/s2" -----> distance = 1.5t^2
a = 3 m/s^2
v = 3t + c
when t = 0 , v = 0 (it started from rest) , so c = 0
v = 3t
s = 1.5^2 + k
when t = 0, s = 0, then k = 0
so s = 1.5t^2
so how far has it gone in 7 s ?
s = 1.5(49) or 73.5 m, with a velocity of 3(7) or 21 m/s
2nd stage:
a2 = -2.5
v2 = -2.5t + c
here c = 21
and s2 = -1.25t^2 + 21t + 0
after 2 seconds of this
s2 = -1.25(4) + 21(2) or 37 m
so the car went 73.5 m in the first stage and another 37 slowing down,
so it went 110.5 m
check my arithmetic
To determine the distance the car has traveled, we need to calculate the distance traveled during each phase separately and then add them together.
Phase 1: Acceleration with a = +3.0 m/s² for t = 7.0 s.
Using the kinematic equation: s = ut + 0.5at², where u is the initial velocity equal to 0,
s1 = 0.5 * 3.0 * (7.0)²
Phase 2: Deceleration with a = -2.5 m/s² for t = 2.0 s.
Using the same kinematic equation, we have:
s2 = 0.5 * (-2.5) * (2.0)²
Now, we can calculate the total distance traveled:
Total distance = s1 + s2
To determine how far the car has gone, you can use the kinematic equation:
s = ut + (1/2)at^2
where:
s = distance
u = initial velocity (in this case, the car starts from rest, so u = 0)
t = time
a = acceleration (in this case, the acceleration is +3.0 m/s^2 for the first 7.0s and -2.5 m/s^2 for the next 2.0s)
For the first phase of motion (uniform acceleration of +3.0 m/s^2 for 7.0s):
s1 = 0 + (1/2)(3.0)(7.0)^2
For the second phase of motion (uniform acceleration of -2.5 m/s^2 for 2.0s):
s2 = 0 + (1/2)(-2.5)(2.0)^2
To find the total distance, we add the distances from both phases:
Total distance = s1 + s2
Now, let's substitute the values into the equations to calculate:
s1 = 0 + (1/2)(3.0)(7.0)^2 = 73.5 m
s2 = 0 + (1/2)(-2.5)(2.0)^2 = -5.0 m
Total distance = 73.5 m + (-5.0 m) = 68.5 m
Therefore, the car has traveled a total distance of 68.5 meters.