Problem 9 – time needed to brake a car

A car can be braked to stop from 100 km/h in 45 m. What is the magnitude of acceleration, assuming it is constant? What is the stopping time?

Solution.

Given are:

v₀ = 100.0 km/h   - initial velocity

v_F = 0.0 km/h       - final velocity

D = 45 m distance traveled

We have to find:

a = ?          – acceleration during braking the car

t = ?           – time of needed to stop the car

 

We write equations for final velocity in motion with constant acceleration and for distance traveled when initial velocity is v₀ and acceleration is a.

v_F = v₀ – at                (1)

D = v₀t – (1/2) at²      (2)

a is negative because its direction is opposite to velocity.

From the Eq. 1 we have

a = v₀ / t                     (3)        because v_F = 0

Now we substitute this expression for a into Eq. 2.

D = v₀ t – (1/2) (v₀ / t) t²

after a little algebra

D = (1/2) v₀ t

from this

t = 2D / v₀      (4)

Substituting Eq. 4 into Eq. 3 we get

a = (1/2)v₀²/ D   (5)

Substituting numbers given in problem into Eqs. 4 and 5 (remember to change km/h into m/s) we get

a = 8.57 m/s²  

t = 3.24 s       

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