Question

For the first 4 seconds, the displacement, in metres, of a sports car from its initial position is given by s=12t^2-t^3. The national speed limit in Great Britain is 70mph. Ath the end of 4 seconds, would the driver be breaking the law?

Answer 1

lol, nice question by the way.

Answer 2

The velocity of the car at time \(t\) is given by \(\dfrac{d}{dt}s\). Let's calculate what it is.

Answer 3

\[v(t) = -3t^2 + 24t\]All righty, then. Now we have to calculate the maximum velocity the driver goes at. What is the max of \(v(t)\)?

Answer 4

I dont understand this equation...\[t=ds/dt\] Could you please explain.

Answer 5

Do you know that the derivative of displacement is velocity?

Answer 6

no

Answer 7

Do you know what derivative, displacement (or distance) and velocity (or speed) are?

Answer 8

I know what they are, i just dont understand the 2 key relationships such as \[v=ds/dt \] and \[a=dv/dt\]

Answer 9

All right. The derivative of something is just the rate-of-change of something. Do you understand that? If so, then it should be very easy to explain.

Answer 10

yes

Answer 11

And you know that,

1. Velocity is the rate of change of displacement.
2. Acceleration is the rate of change of velocity.

Answer 12

yes

Answer 13

Thus, we can conclude:

1. Velocity is the *derivative* of displacement.
2. Acceleration is the *derivative* of velocity.

Got it? :D

Answer 14

So , to work out the velocity, you differentiate the displacement

Answer 15

Exactly.

Similarly, for acceleration, you differentiate velocity.

Answer 16

got it...

Answer 17

So now could you answer the question plz...

Answer 18

So what do you do after you differentiate the displacement getting \[-3t^2+24\]

Answer 19

Great stuff! :)

So \(v(t) = \dfrac{d}{dt}s(t) = -3t^2 + 24t\).

You need to figure out whether \(-3t^2 + 24t\) goes above \(70\) in the interval \(0\le t \le 4\) because the displacement is given for the first four seconds.

Answer 20

okay

Answer 21

\(-3t^2 + 24t\) is the velocity, and the guy gets arrested if this function goes over 70.

Can you figure out the maximum value of \(-3t^2 + 24\) in the interval \([0,4]\)?

Answer 22

-24

Answer 23

I mean 48

Answer 24

Eek, I forgot one more thing: there is a difference between speed and velocity. Velocity is the absolute value of speed.

Answer 25

Speed is the absolute value of velocity**

Answer 26

Meaning that no matter what the velocity is, the speed will always be positive.

If velocity = -24, then speed = 24.

Answer 27

So we have to find the minimum of the function.

Answer 28

If for example, the minimum velocity is -80, then the speed would be 80 which still exceeds the max speed.

Answer 29

okay

Answer 30

So what would the next step be...

Answer 31

i Have worked out the answer. it is \[172.8 km h ^{-1}\], as i got the answer of \[48ms ^{-1}\] from differentiate the displacement and putting 4.
Then i had to convert that number to miles per hour and so it was \[\frac{ 48 \times 60 \times 60 }{ 1000 }\], which equaled 172.8. So the person was faster than the national speed limit.