Using derivatives: A car’s position is given by s = 25t + t^2

[Amr]

Using derivatives: A car’s position is given by s = 25t + t^2

A car’s position is given bys = 25t + t^2where s is in metres and t is in seconds. The car passesa police car that is travelling at 20 m/s. The police officer turns on the siren and begins to accelerateat 1.5 m/s^2 to chase the speeding car. The velocity of the police car is given byv =1.5t + 20
a) When is the speeding car travelling at 31 m/s?
b) How fast is the police car travelling at that time?
c) How far apart are the two cars at that time?
d) Will the police car ever catch up to the speeder?

For c), how would you find the displacement of the cop car?
and for d), is an inequality used?

 

[HallsofIvy]

A car’s position is given bys = 25t + t^2where s is in metres and t is in seconds. The car passesa police car that is travelling at 20 m/s. The police officer turns on the siren and begins to accelerateat 1.5 m/s^2 to chase the speeding car. The velocity of the police car is given byv =1.5t + 20
a) When is the speeding car travelling at 31 m/s?

The position of the speeding car is given by s= 25t+ t^2 and its speed is the derivative of the positon function. The speed s given by 25+ 2t. I presume, since you are not asking about this question, you did that and solved 25+ 2t= 31 to get t= 3.

b) How fast is the police car travelling at that time?

And, of course, you then put t= 3 into 1.5t+ 20, to get 4.5+ 20= 24.5

c) How far apart are the two cars at that time?
Contrary to the title of this thread, [this] is not a matter of using the derivative (you used that in (a)) but rather the anti-derivative. If the speed of the police car is given 1.5t+ 20, its position is given by (1.5/2)t^2+ 20t= 7.25t^2+ 20t. The "constant of integration" is 0 since the position function for the speeding car gives 0 as t passes the police car. The distance between the two cars (the speeding car is ahead of course) is 25t+ t^2- (7.25t^2+ 20t)= 5t- 6.25t^2

d) Will the police car ever catch up to the speeder?

Is 5t- 6.25t^2 ever equal to 0 for a positive t?

For c), how would you find the displacement of the cop car?
and for d), is an inequality used?