Reaction time while stopping a car

In summary: Step 1: use a formula that doesn't involve time, to get v1 in terms of a and to. Step 2: use a formula that doesn't involve v1 to get a and to in terms of s.In summary, the problem involves determining the reaction time and deceleration of a car as it slows down under constant braking. The total distance traveled during both the reaction time and braking phases is given for two different initial speeds. Using the kinematic equations and treating the motion as having two phases, we can set up two equations to solve for the reaction time and deceleration. The assumption is made that the total time for all the effects of the reaction time is constant and
  • #1
baldbrain
236
21

Homework Statement


To stop a car, first you require a certain reaction time to begin braking; then the car slows down under constant breaking decelaration. Suppose that the total distance moved by the car during these two phases is 56.7 m when its initial speed was 80.5 km/h and 24.4 m when its initial speed 48.3 km/h. What are
(a) The reaction time
(b) Magnitude of decelaration

2. Homework Equations

The Kinematical Equations

The Attempt at a Solution


Actually, my guess of what they're asking is the time between pressing the brake pedal and the brake pads squeezing in on the tyres.
So let to be the reaction time and a be the constant breaking decelaration.
u1 = 80.5 km/h = 22.36 m/s
u2 = 48.3 km/h = 13.42 m/s
Now, I thought of using s = ut + (1/2)at2, get two equations and solve them, but I have one basic question.
In the time interval I think they're asking, the car isn't decelarating. It starts decelarating after the brake pads squeeze in on the tyres. So how can I substitute to & a in the same equation?
 
Physics news on Phys.org
  • #2
Treat the "reaction time" period as having constant velocity. (The acceleration hasn't begun yet.) After the brakes are applied (and engaged), treat it as having constant acceleration.
 
  • #3
56.7 = (22.36)(t + to) - (1/2)a(t2)
24.4 = (13.42)(t + to) - (1/2)a(t2)

Assuming t is the braking time and to is the reaction time.
3 variables, 2 equations...
 
  • #4
baldbrain said:
56.7 = (22.36)(t + to) - (1/2)a(t2)
24.4 = (13.42)(t + to) - (1/2)a(t2)

3 variables, 2 equations...
Assuming t is the braking time and to is the reaction time.
Treat the motion as having two phases. For the accelerated phase, use a different kinematic formula that doesn't call for t.
 
  • #5
Doc Al said:
For the accelerated phase, use a different kinematic formula that doesn't call for t.
There's only v2 = u2 - 2as. But v will be zero in each case. So I will have two inconsistent values of a.
 
  • #6
baldbrain said:

Homework Statement


To stop a car, first you require a certain reaction time to begin braking; then the car slows down under constant breaking decelaration. Suppose that the total distance moved by the car during these two phases is 56.7 m when its initial speed was 80.5 km/h and 24.4 m when its initial speed 48.3 km/h. What are
(a) The reaction time
(b) Magnitude of decelaration

2. Homework Equations

The Kinematical Equations

The Attempt at a Solution


Actually, my guess of what they're asking is the time between pressing the brake pedal and the brake pads squeezing in on the tyres.
So let to be the reaction time and a be the constant breaking decelaration.
u1 = 80.5 km/h = 22.36 m/s
u2 = 48.3 km/h = 13.42 m/s
Now, I thought of using s = ut + (1/2)at2, get two equations and solve them, but I have one basic question.
In the time interval I think they're asking, the car isn't decelerating. It starts decelerating after the brake pads squeeze in on the tyres. So how can I substitute to & a in the same equation?

In this problem, "reaction time" is a sum: the time it takes for the foot to rise from the car floor to the brake pedal, plus the time needed to depress the pedal, plus the time you mentioned above. The assumption is that the total time for all of those effects is not affected by the initial speed, and that the deceleration itself is likewise independent of the initial speed. (That last one seems suspicious to me, as it assumes we push equally hard on the brake pedal whether we are slowing down in a controlled way before a red light or whether we are desperately trying to avoid a rear-end collision.)
 
  • #7
baldbrain said:
There's only v2 = u2 - 2as. But v will be zero in each case. So I will have two inconsistent values of a.
The final velocity is zero, of course. But the initial velocity and distance will be different in each case.
 
  • #8
Ray Vickson said:
(That last one seems suspicious to me, as it assumes we push equally hard on the brake pedal whether we are slowing down in a controlled way before a red light or whether we are desperately trying to avoid a rear-end collision.)
Yeah, that's quite ambiguous
 
  • #9
@Ray Vickson is correct. But ignore those real-life details (:wink:) and solve the idealized problem: Treat the reaction time and acceleration as constants to be determined.
 
  • #10
Doc Al said:
The final velocity is zero, of course. But the initial velocity and distance will be different in each case.
So, you mean a formula with distance, decelaration and initial velocity that doesn't include braking time but includes the reaction time? Then, I don't get what you're trying to say?
 
  • #11
baldbrain said:
So, you mean a formula with distance, decelaration and initial velocity that doesn't include braking time but includes the reaction time? Then, I don't get what you're trying to say?
Don't try to jam it into one equation. (At least not at first.)

Phase 1: Constant speed for reaction time (Distance = D1)
Phase 2: You have the correct formula, which has distance, acceleration, and velocity (Distance = D2)

(You are given the total distance, D1 + D2.)
 
  • #12
Doc Al said:
You are given the total distance, D1 + D2.
The equations in #3 do incorporate D1 + D2. That's why there's t + to in the first term but only t2 in the second.
By the way, if you subtract Eq. 1 and 2, you can obtain total time T, which comes out to be 3.61 s. Maybe we could use that.
 
  • #13
baldbrain said:
The equations in #3 do incorporate D1 + D2. That's why there's t + to in the first term but only t2 in the second
I would skip those equations and use one that doesn't require t, which is an unknown. Like you had in #5.
 
  • #14
I'm thinking of substitutimg t = u/a in Eq. 1 & 2, but that t + to term complicates it.
 
  • #15
baldbrain said:
I'm thinking of substitutimg t = u/a in Eq. 1 & 2, but that t + to term complicates it.
Once again, why not just use the equation relating distance, acceleration, and speed? Then you won't have to go through the pain of eliminating the unnecessary time unknown.
 
  • #16
But, that won't render anything... v is zero in both cases. I'll get 2 inconsistent values of a
 
  • #17
baldbrain said:
But, that won't render anything... v is zero in both cases. I'll get 2 inconsistent values of a
The final velocity is zero! So what? You still have the initial velocity!
 
  • #18
Doc Al said:
The final velocity is zero! So what? You still have the initial velocity!
u12 = 2as1
u22 = 2as2
If you substitute the given values, you get 2 different values of a.
 
  • #19
##0 = u^2 - 2as## is valid only in the case of constant acceleration. This is only one of the phases of each of the situations. You are completely missing the reaction phase by just plugging the values into this equation.
 
  • Like
Likes baldbrain
  • #20
baldbrain said:
u12 = 2as1
u22 = 2as2
If you substitute the given values, you get 2 different values of a.
That equation is only for the 2nd phase of the motion. Don't neglect the first phase. (The distance in that equation is not the total distance that was given in the problem statement.)
 
  • Like
Likes baldbrain
  • #21
I'm still stuck at t + to. If only we could substitute t= u/a in t2 and that t in t + to would just magically vanish...
 
  • #22
Things do not happen by magic in physics, they happen by logical reasoning given the assumed situation. You need to express the ##s## that should go into the equations in terms of the reaction time ##t_0## and the initial velocity ##v_i## and the total stopping distance ##s_0##. Once you have done that, you will have two equations with two unknowns (##t_0## and ##a##) which is a perfectly solvable system.
 
  • #23
Orodruin said:
##0 = u^2 - 2as## is valid only in the case of constant acceleration. This is only one of the phases of each of the situations. You are completely missing the reaction phase by just plugging the values into this equation.
Orodruin said:
You need to express the ##s## that should go into the equations in terms of the reaction time ##t_0## and the initial velocity ##v_i## and the total stopping distance ##s_0##. Once you have done that, you will have two equations with two unknowns (##t_0## and ##a##) which is a perfectly solvable system.
Doc Al said:
That equation is only for the 2nd phase of the motion. Don't neglect the first phase. (The distance in that equation is not the total distance that was given in the problem statement.)
Yes, I get what you're saying mow. I was being an ass...
56.7 m = d11 + d12
= u1to + u12/2a
24.4 m = d21 + d22
= u2to + u22/2a
 
  • #24
I'm getting to = 0.74 s and a = 6.23 m/s2.
 
  • #25
Thank you all. And apologies for behaving rather idiotically
 
  • #26
baldbrain said:
I'm getting to = 0.74 s and a = 6.23 m/s2.

This is not quite right. You need to be more precise with the numbers.
 
  • #27
verty said:
This is not quite right. You need to be more precise with the numbers.
2-decimal accuracy is more than enough...
Are you getting a different answer?
The values don't satisfy the equation?
 
  • #28
baldbrain said:
2-decimal accuracy is more than enough...
That is not what he is saying, he is saying you have a rounding error.
 
  • #29
Orodruin said:
That is not what he is saying, he is saying you have a rounding error.
I did them using a calculator
a = 6.23724686 m/s2
to = 0.74098462 s
6.23 or 6.24 hardly matters in my case.
 
  • #30
baldbrain said:
I did them using a calculator
a = 6.23724686 m/s2
to = 0.74098462 s
6.23 or 6.24 hardly matters in my case.
You are given data to three significant figures. Standard is that you quote an answer to the same, so it would seem that you should give 62.4 as the answer.
But when the calculation involves taking differences this can go wrong. In the present case, replacing the 24.4 with 24.44 gives an acceleration of 6.26 m/s2. So the really correct way to answer would be something like 6.24±0.02.
 
  • Like
Likes baldbrain

Related to Reaction time while stopping a car

1. What is reaction time while stopping a car?

Reaction time while stopping a car is the amount of time it takes for a driver to react and apply the brakes after perceiving a potential hazard on the road.

2. How is reaction time while stopping a car measured?

Reaction time while stopping a car can be measured by recording the time it takes for a driver to perceive a hazard and apply the brakes, using specialized equipment such as a driving simulator or through real-world driving tests.

3. What factors can affect reaction time while stopping a car?

There are several factors that can affect reaction time while stopping a car, including the driver's age, fatigue, distractions, alcohol or drug use, road conditions, and vehicle speed.

4. How can we improve reaction time while stopping a car?

There are a few ways to improve reaction time while stopping a car, such as practicing defensive driving techniques, getting enough rest before driving, avoiding distractions, and maintaining a safe distance from other vehicles on the road.

5. Why is reaction time while stopping a car important?

Reaction time while stopping a car is important because it can determine how quickly a driver can respond to potential hazards on the road, potentially preventing accidents and ensuring the safety of the driver and others on the road.

Similar threads

  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
5K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
33
Views
3K
  • Introductory Physics Homework Help
Replies
13
Views
861
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
531
  • Introductory Physics Homework Help
Replies
1
Views
1K
Back
Top