A thief in a stolen car passes through a police check post at his top speed of 90 km/h. A motorcycle cop, reacting after 2 s, accelerates from rest at 5 ms. His top speed being 108 km/h. At what distance from checkpost they will be caught?

To determine the distance from the check post where the motorcycle cop catches the thief, we need to analyze the motion of both the thief and the cop. We'll break this down step by step, considering their speeds, accelerations, and the time it takes for the cop to react.

Understanding the Thief's Motion

The thief is traveling at a constant speed of 90 km/h. To work with this speed in meters per second (m/s), we convert it:

• 90 km/h = 90 × (1000 m / 1 km) × (1 h / 3600 s) = 25 m/s

Since the thief is moving at a constant speed, the distance he covers in time \( t \) can be expressed as:

Distance (thief) = Speed × Time

Analyzing the Cop's Motion

The motorcycle cop starts from rest and accelerates at 5 m/s². However, he has a reaction time of 2 seconds before he starts moving. During this time, the thief continues to travel. The distance the thief covers during the cop's reaction time is:

Distance (thief during reaction time) = Speed × Time = 25 m/s × 2 s = 50 m

Calculating the Cop's Acceleration Phase

After the 2 seconds, the cop begins to accelerate. His acceleration is 5 m/s², and he will continue to accelerate until he reaches his top speed of 108 km/h. Converting this speed to m/s gives us:

• 108 km/h = 108 × (1000 m / 1 km) × (1 h / 3600 s) = 30 m/s

Now, we need to find out how long it takes the cop to reach this top speed. Using the formula:

Final Speed = Initial Speed + (Acceleration × Time)

Since the initial speed is 0, we can rearrange this to find the time:

Time = Final Speed / Acceleration = 30 m/s / 5 m/s² = 6 s

Distance Covered by the Cop During Acceleration

Next, we calculate the distance the cop covers while accelerating. The formula for distance under constant acceleration is:

Distance = Initial Speed × Time + (1/2) × Acceleration × Time²

Since the initial speed is 0, this simplifies to:

Distance = (1/2) × 5 m/s² × (6 s)² = (1/2) × 5 × 36 = 90 m

Finding the Total Distance and Catch-Up Point

Now, we can find the total distance the thief has traveled when the cop reaches his top speed. The total time from the start until the cop reaches his top speed is:

Total Time = Reaction Time + Acceleration Time = 2 s + 6 s = 8 s

During this total time, the thief travels:

Distance (thief) = Speed × Total Time = 25 m/s × 8 s = 200 m

At this point, the thief has traveled 200 meters, while the cop has traveled 50 meters during the reaction time and 90 meters during the acceleration phase, totaling:

Total Distance (cop) = 90 m

Determining the Catch-Up Distance

After 8 seconds, the thief is 200 meters away, and the cop is 90 meters away. The distance between them is:

Distance between them = 200 m - 90 m = 110 m

Now, the cop is traveling at 30 m/s, while the thief continues at 25 m/s. The relative speed at which the cop is closing the gap is:

Relative Speed = Speed (cop) - Speed (thief) = 30 m/s - 25 m/s = 5 m/s

To find out how long it takes for the cop to close the 110-meter gap:

Time to catch up = Distance / Relative Speed = 110 m / 5 m/s = 22 s

Final Distance from the Check Post

During this additional 22 seconds, the thief continues to travel:

Distance (thief) = Speed × Time = 25 m/s × 22 s = 550 m

Adding this to the initial distance of 200 meters gives us:

Total Distance (thief) = 200 m + 550 m = 750 m

Thus, the cop catches the thief at a distance of 750 meters from the check post.