So I'm in intro to mechanics at the moment, and I'm fairly rusty on most of this. Was hoping someone might be able to offer me some guidance on a problem I'm snagged on:

A vehicle from rest starts accelerating at a constant rate of 2.2 m/s^2, until it reaches a speed of 27 m/s. At that point, it begins to slow at a constant rate of 1.0 m/s^2, until it stops. (a) how much time elapses from start to stop? (b) How far does the vehicle move from start to stop?

the equations I have to work with are these:

term = ending, init = initial (applies to position and velocity)
A = acceleration
X = position
V = velocity
T = time

(1) X_term = X_init + V_init * T + .5AT^2
(2) V_term = V_init + AT
(3) V^2 = V_init^2 + 2A(X_term-X_init)

I read this as two separate sets of solutions, with the second (when the car starts to break) being dependent on the solutions to the first (when the car is accelerating).

The way I have it set up is as follows:

Problem 1 (car accelerating)
X_init1 = 0
X_term1 = ?
T1 = ?
V_init1 = 0 m/s
V_term1 = 27 m/s
A = 2.2 m/s^2

Solution: Use equation 2 to get T1, use T1 in equation 1 to get X_term1.

Problem 2 (car braking)
X_init2 = X_term1
X_term2 = ?
T2 = ?
V_init2 = 27 m/s
V_term2 = 0 m/s
A = -1.0 m/s^2

Solution: Use equation 2 to get T2, use T2 in equation 1 to get X_term2.

I'm confident that when I do this, I need to sum T1 and T2 (to get time elapsed from start to finish), but will I need to sum X_term2 and X_term1 as well to get the distance covered? My thinking was no, if I'm using the initial position that is the terminal position of the first problem this should already be taken into account.

Sorry if this seems like a silly question, just want to make sure I'm not thinking about this the wrong way.

The problem is simpler than it may sound from general principle (although that's still pretty simple) because you have two constant accelerations and a zero in each velocity curve (line, really). The easy way to do it is:

1) divide the known final velocities for each segment by their acceleration to determine the time required to reach that
2) compute the average velocity across that period (a useful predicate of the Mean Value Theorem because the derivative is the same for the entire segment) by dividing the final by 2
3) multiply result result of step 2 (average velocity) by step 1 (time in motion)

Repeat for the deceleration phase except you use the starting velocity instead of the end (which is zero), and sum the answers.


Does this course assume you have taken basic collegiate calculus?

The requirement is Calc 1 (possibly 2, I forget). I took it a while ago so I'm rather rusty on the mathematics, but its all floating around in my head and will come back to me once I've knocked the dust off. Its more the logic behind setting up the solutions that I'm not quite comfortable with yet.

I'm told that the equations that I listed will become less useful when integrals and derivatives come into play, but for now those are what we're supposed to be working with.

So I got the answer to that one, and it turns out that was the easy question of the bunch.

To stop a car, first you require a certain reaction time to begin braking; then the car slows under the constant braking deceleration. Suppose that the total distance moved by your car during these two phases is 58.4 m when its initial speed is 76.8 km/h, and 26.5 m when its initial speed is 47.4 km/h. What are (a) your reaction time and (b) the magnitude of the deceleration?

Same set of equations to use as listed before. I thought that I'd have to set up simultaneous equations to solve for the unknowns necessary, but I have no idea where to go from there. Any pointers on where to get started?

So I got the answer to that one, and it turns out that was the easy question of the bunch.

To stop a car, first you require a certain reaction time to begin braking; then the car slows under the constant braking deceleration. Suppose that the total distance moved by your car during these two phases is 58.4 m when its initial speed is 76.8 km/h, and 26.5 m when its initial speed is 47.4 km/h. What are (a) your reaction time and (b) the magnitude of the deceleration?

Same set of equations to use as listed before. I thought that I'd have to set up simultaneous equations to solve for the unknowns necessary, but I have no idea where to go from there. Any pointers on where to get started?
I'm assuming that the reaction time and deceleration is the same for both, otherwise this isnt solvable into a discrete answer. Therefore you need to solve for reaction time, then you can get deceleration.

Basically,
x_traveled = (distance traveled before reacting) + (distance traveled while decelerating)
x_traveled = v_car * reaction_time + (v_car * time_to_stop + 1/2 * deceleration * time_to_stop^2)

It's cumbersome to write out the whole thing on the forum, but you end up with (you need to substitute in time_to_stop = v_car / deceleration):

x_traveled = v_car * reaction_time + (3 * v_car^2) / (2 * deceleration)

You know x_traveled and v_car, which leaves you with a 2 variable equation. Solve the above equation for reaction_time or deceleration (your choice), then you have 2 sets of data that you can now use to solve for it.

i.e.:

reaction_time = (x_traveled - (3 * v_car^2) / 2 * deceleration) / v_car

There's a few ways you can go and solve this with the data. One way is to set the equation equal to itself, plug in constants, and solve:

(x_traveled - (3 * v_car1^2) / 2 * deceleration) / v_car1 = (x_traveled - (3 * v_car2^2) / 2 * deceleration) / v_car2

This is just off the top of my head, typing it out on a forum, so you may want to check the algebra.. but the process should be sound.