Question
The Helena moves at \(5\text{ m s}^{-1}\) and is already \(200\text{ m}\) from the dock.
A smaller boat leaves the dock from rest and accelerates at \(0.5\text{ m s}^{-2}\).
Find the distance from the dock at which the smaller boat catches the Helena.
Try the question yourself first. If you get stuck, open the hints before using the full walkthrough.
Find a displacement equation for each boat, set them equal, solve for the time, then substitute back to find the distance from the dock.
Step 1
Write the Helena's displacement equation
Let \(t\) be the number of seconds after the smaller boat leaves the dock. Which displacement model matches the Helena?
Step 2
Write the smaller boat's displacement equation
The smaller boat starts from the dock at rest and accelerates at \(0.5\text{ m s}^{-2}\). Which displacement equation is correct?
Step 3
Set the displacements equal
At the moment the smaller boat catches the Helena, both boats are at the same displacement.
Step 4
Solve the quadratic for time
Type the two times that solve the catch-up equation, separated by a comma.
Step 5
Choose the physical solution
Which time should we use in the context of the question?
Step 6
Find the distance from the dock
Substitute \(t=40\) into either displacement equation.
Step 7
Final answer
Now answer the question in context.
\[
s_H=200+5t
\qquad
s_B=0.25t^2
\]
\[
0.25t^2=200+5t
\]
\[
t^2-20t-800=0
\]
\[
(t-40)(t+20)=0
\]
\[
t=40 \text{ s}
\]
\[
s=200+5(40)=400
\]
Great work. The smaller boat catches the Helena 400 m from the dock.