Question

Q: A child is running at her maximum speed of 4.0 m/s to catch an ice-cream truck, which is stopped at the side of the road. When the child is 20 m from the truck, the ice-cream truck starts to accelerate away at a rate of 1.0 m/s² . Does the child catch the truck? solve this problem algebraically and graphically.

Answer 1

Algebraic method :
Let the velocity of the child is v = 4.0 m/s. And the acceleration of the ice cream truck is a = 1.0 m/s^2.
And as there is an initial gap d = 20 m, so, we have at time t, the distance travelled by the child is
  
And similarly, the distance travelled by the ice cream truck is
   
And the child will be able to catch the truck if at the time t the child also cover the extra initial distance between the child and the truck. So, the condition is
   
  
  And so, we get
  
  

Now in the given case,
     
And so,
   
So, the time is imaginary. So, this implies that the child will never be able to catch the truck.

Graphically :
  We plot the two functions
   
And
   
on the same plot. And this implies
  
And the two plots implies that they will not intersect at any finite time t. And so, we see that the child will not be able to catch the truck.