Braking, stopping time and stopping distance — 9709 P4
Use a negative acceleration for a vehicle still moving forwards.
Syllabus 4.2
Notes and key methods
For braking from forward speed u with constant deceleration magnitude b, take \(\displaystyle a=-b\). At the first stop, \(\displaystyle v=0\). The stopping time and braking distance follow from the relations below.
The model ends at the first stop unless continued motion is explicitly specified. Continuing \(\displaystyle a=-b\) beyond that time would predict reversal, not continued braking to rest. With a reaction interval, add the distance at the unchanged speed before braking begins.
Choose and retain a positive direction; distinguish signed quantities from magnitudes.
Use SI units and \(\displaystyle g= \frac{10m}{s^{2}}\) wherever gravity is relevant.
Apply only the model and time interval stated in the question.
Three worked examples
Example 1 · Braking, stopping time and stopping distance
A vehicle travels forwards at \(\displaystyle \frac{13m}{s}\). After a reaction interval of \(\displaystyle \frac{1}{2}\) s at unchanged speed, it brakes with constant deceleration magnitude \(\displaystyle \frac{2m}{s^{2}}\) until its first stop. Ignore all other effects.
Find the braking time in s, excluding reaction time.
Find the braking distance in m, excluding reaction distance.
Find the reaction distance in m.
Find the full stopping distance from the start of the reaction interval, in m.
Show the reasoning
(a)
Apply the zero final velocity to the braking stage only.
Set 0=u−bt to obtain \(\displaystyle \frac{13}{2}\) s.
Answer: \(\displaystyle \frac{13}{2}\).
(b)
Use the signed braking acceleration in the time-free equation.
Using 0=u²−2bD gives \(\displaystyle \frac{169}{4}\) m.
Answer: \(\displaystyle \frac{169}{4}\).
(c)
Use constant speed during the stated reaction interval.
Unchanged speed times reaction time gives \(\displaystyle \frac{13}{2}\) m.
Answer: \(\displaystyle \frac{13}{2}\).
(d)
Combine the two stage distances without including a post-stop reversal.
Add the reaction and braking distances to obtain \(\displaystyle \frac{195}{4}\) m.
Answer: \(\displaystyle \frac{195}{4}\).
Example 2 · Braking, stopping time and stopping distance
A vehicle travels forwards at \(\displaystyle \frac{14m}{s}\). After a reaction interval of \(1\) s at unchanged speed, it brakes with constant deceleration magnitude \(\displaystyle \frac{3m}{s^{2}}\) until its first stop. Ignore all other effects.
Find the braking time in s, excluding reaction time.
Find the braking distance in m, excluding reaction distance.
Find the reaction distance in m.
Find the full stopping distance from the start of the reaction interval, in m.
Show the reasoning
(a)
Apply the zero final velocity to the braking stage only.
Set 0=u−bt to obtain \(\displaystyle \frac{14}{3}\) s.
Answer: \(\displaystyle \frac{14}{3}\).
(b)
Use the signed braking acceleration in the time-free equation.
Using 0=u²−2bD gives \(\displaystyle \frac{98}{3}\) m.
Answer: \(\displaystyle \frac{98}{3}\).
(c)
Use constant speed during the stated reaction interval.
Unchanged speed times reaction time gives \(14\) m.
Answer: \(14\).
(d)
Combine the two stage distances without including a post-stop reversal.
Add the reaction and braking distances to obtain \(\displaystyle \frac{140}{3}\) m.
Answer: \(\displaystyle \frac{140}{3}\).
Example 3 · Braking, stopping time and stopping distance
A vehicle travels forwards at \(\displaystyle \frac{15m}{s}\). After a reaction interval of \(\displaystyle \frac{3}{2}\) s at unchanged speed, it brakes with constant deceleration magnitude \(\displaystyle \frac{4m}{s^{2}}\) until its first stop. Ignore all other effects.
Find the braking time in s, excluding reaction time.
Find the braking distance in m, excluding reaction distance.
Find the reaction distance in m.
Find the full stopping distance from the start of the reaction interval, in m.
Show the reasoning
(a)
Apply the zero final velocity to the braking stage only.
Set 0=u−bt to obtain \(\displaystyle \frac{15}{4}\) s.
Answer: \(\displaystyle \frac{15}{4}\).
(b)
Use the signed braking acceleration in the time-free equation.
Using 0=u²−2bD gives \(\displaystyle \frac{225}{8}\) m.
Answer: \(\displaystyle \frac{225}{8}\).
(c)
Use constant speed during the stated reaction interval.
Unchanged speed times reaction time gives \(\displaystyle \frac{45}{2}\) m.
Answer: \(\displaystyle \frac{45}{2}\).
(d)
Combine the two stage distances without including a post-stop reversal.
Add the reaction and braking distances to obtain \(\displaystyle \frac{405}{8}\) m.
Answer: \(\displaystyle \frac{405}{8}\).
Common mistakes
Do not substitute a positive b as signed acceleration in a forward positive axis.
Separate the reaction interval from the braking interval.