Gregory L. answered • 03/07/21
Physics and math help! Super passionate
Since this problem deals only with how the car is moving, and not why (i.e. there is no mention of forces) this means we are dealing with the subject of kinematics. Another clue from the problem that we can use kinematics is that it specifies uniform acceleration.
Let's start by remembering the big four kinematic equations and what the symbols mean:
1) d = vi + (1/2)at2
2) vf2 = vi2 + 2ad
3) vf = vi + at
4) d = [(vi+ vf) / 2]*t
Where d is the distance, vi & vf are the initial and final velocity, a is the acceleration, and t is the time given.
Looking at the problem, what are the values we are given and what are we trying to solve for?
- The car is at rest initially, so vi = 0.
- The time of acceleration is given at t = 5.21.
- The distance traveled is d = 110m.
Part A of the question asks to determine the acceleration, and part B asks us to determine the final velocity.
So which of the kinematic equations should we use to solve for and why? One helpful tip in determining which one you should use is to remember that with 1 equation you can only ever solve for 1 unknown quantity. Equations 2 & 3 both feature 2 of our unknown quantities, vf & a so we cannot use them to start out.
Equations 1 and 4 look promising. We can use 1 to solve for a, and we can use 4 to solve for vf. Let's start with 1.
1) d = vi + (1/2)at2
If we rearrange this using algebra we obtain:
a = 2(d - vi) ⁄ t2
And plugging in the values we are given we see that:
a = 2(110m - 0) / (5.21s)2
a = 8.10 m/s2
Now we have some options. Any of the kinematic equations 2, 3, or 4 are available to use to find vf so it's a matter of choice how we want to proceed. I will work through all 3 and review the advantages of each.
2) vf2 = vi2 + 2ad
This looks easy enough, let's plug in what we know and crank the gears:
vf2 = (0 m/s)2 + 2(8.10 m/s2) * (110m)
vf2 = 1782 m2/s2
vf = 42 m/s
3) vf = vi + at
Straight forward...
vf = (0m/s) + (8.10 m/s2) * (5.21s)
vf = 42 m/s
4) d = [(vi+ vf) / 2]*t
This is the trickiest one to use. Notice that it deals with the quantity (vi+ vf) / 2 - this is an expression for the average velocity, and this equation makes it clear why theses 4 equations only work with uniform acceleration. So rearranging the equation a bit:
vf = 2 * (d / t) - vi
vf = 2*(110m / 5.21s) - 0m/s
Vf = 42 m/s
As you can see, all options lead to the same answer. Which equation 2-4 you use to solve for the final velocity is up to you and your comfort level using algebra. #3 I would say is the most plug and play, then followed by #2 and #4 in terms of usability.
Gregory L.
I'm sorry I made a typo that luckily doesn't actually affect the problem. Equation 1) is actually d = vi * t + (1/2)at^2. In this problem since the initial velocity is 0 it doesn't make a difference, but a very important typo.03/07/21