Question

Solve Differential Equation

The skid marks made by an automobile indicates that its breaks were first applied at a distance of 60 meters before it came to a complete stop. The car is known to have a constant deceleration of 20 m/s2 under these conditions.
How fast (in km/h) was the car traveling when the brakes were first applied?

Please sketch coordinate system to support answer

Answer 1

Step 1)

As given the kid marks made by an automobile indicates that its breaks were first applied at a distance of 60 meters before it came to a complete stop. The car is known to have a constant deceleration of 20 m/s² under these conditions.

at t = 0 we have distance traveled x = 0 and initial velocity v = unknown value it means we can write,

at t = t we have distance traveled x = 60 meters and final velocity v = 0 it means we can write,

we can sketch the situation as below :

As deceleration is constant throughout we can write,

minus sign indicates deceleration

we know that velocity is integral of acceleration we can write,

-------------------------------------------------1)

Step 2)

we know that position is integral of the velocity we can write,

-------------------------------------------------2)

we have x₀ = 0 hence put t = 0 and x(0) = 0 in equation 2) we can write,

Put D = 0 in equation 2) we can write,

------------------------------------------------------3)

From equation 1) we have,

At t = t we have v_t = 0 hence put t = t and v(t) = 0 in above equation we can write,

----------------------------------------------------------4)

Put C = 20t in equation 3) we can write,

At t = t we have x_t = 60 hence put t = t and x(t) = 60 in above equation we can write,

but time cannot be negative hence we have,

Step 3)

we have,

when brake were first applied we have t = 0 hence we can write,

It means we can write,

From equation 4) we have,

Hence,

Put t = sqrt(6) we can write,

now convert m/s to km/h we can write,

rounding to two decimal places we can say that,

Hence we can say that car was travelling at a speed of 176.36 km/h when brakes were first applied