A boat is travelling across the water at 18 km/hr due E. A policeman needs to intercept the boat as quickly as possible but is 100 m due S of the boat at t=0 s. If the policeman is initially at rest, and can accelerate his jet ski at 1.0 m/s^2 , what angle must he travel at to intercept the boat in the shortest possible time?
To find the angle at which the policeman should travel to intercept the boat in the shortest possible time, we can use relative motion and the concept of minimizing the time of interception.
Let's break down the problem and analyze the situation:
1. Boat speed and direction:
- The boat is traveling at a constant speed of 18 km/hr (which is equivalent to 5 m/s) due east.
- We will consider the boat's velocity vector as Vb = 5 m/s due east.
2. Initial positions:
- The policeman is initially at rest, and he starts 100 m due south of the boat at t = 0 s.
- We can consider the initial position of the police officer as the origin (0,0) in a cartesian coordinate system.
3. Acceleration of the policeman:
- The policeman can accelerate his jet ski at a constant rate of 1.0 m/s^2.
Now, let's move on to solving the problem:
1. Find the time it takes for the policeman to intercept the boat:
- Let's assume it takes t seconds for the police officer to intercept the boat.
- In this time, the boat will travel a distance of s = Vb * t.
2. Calculate the position of the boat and the policeman after time t:
- The final position of the boat would be at (s, 0) due east, considering it started at (0,0).
- The position of the police officer after time t would be (0, -Vp * t), considering he started at (0,0).
3. Determine the intercept point:
- To intercept the boat, the police officer should reach the same x-coordinate as the boat (s), but with a negative y-coordinate (-s).
- Therefore, the intercept point would be (s, -s).
4. Determine the equation of motion for the police officer:
- The position of the police officer can be represented as Pp = (0, -Vp * t + 0.5 * ap * t^2), where ap is the acceleration of the police officer.
- The initial position of the police officer is (0,0), so we can eliminate the constant term.
5. Equation for the intersecting point:
- Set the x-coordinate of the boat and the police officer equal to each other, s = 0.
=> 0 = 0.5 * ap * t^2 (Equation 1)
- Set the y-coordinate of the boat and the police officer equal to each other, -s = -Vp * t.
=> -s = -Vp * t (Equation 2)
6. Solve for t and ap:
- From Equation 1, we have 0 = 0.5 * ap * t^2.
=> ap = 0 / (0.5 * t^2)
=> ap = 0
- From Equation 2, we have -s = -Vp * t.
=> Vp * t = s
=> Vp = s / t
7. Determine the angle of the police officer's velocity vector:
- The velocity vector of the police officer is in the direction connecting the origin (0,0) and the intercept point (s, -s).
- The angle of the velocity vector can be found using the tangent function:
tan(theta) = (-s / s) (since tan(theta) = opposite/adjacent)
= -1
- Therefore, theta = arctan(-1) = -45 degrees (or 225 degrees in counterclockwise direction from the positive x-axis).
Hence, the police officer must travel at an angle of -45 degrees (or 225 degrees counterclockwise from the positive x-axis) to intercept the boat in the shortest possible time.