A speeding motorist travels with a constant speed of 15 m/s on a straight road. At time t = 0, he passes a hidden police car which is at rest. The instant that the motorist passes, the police car begins accelerating at a constant rate of 2 m/s2 in order to catch the motorist.
a)How many seconds pass before the police car reaches the motorist?
b)How fast is the police car moving when it reaches the motorist?
c) How much distance has the police car traveled when it reaches the motorist?
car location x = 0 + 15 t
police x = (1/2) 2 t^2
so
15 t = t^2
t = 0 (of course) and t = 15 seconds
v = a t = 2 * 15 = 30 m/s
x = (1/2)2 (225) = 225 meters
or since the car went the same distance as the police car
x = 15 m/s * 15 s = 225 meters
To solve this problem, we need to analyze the motion of both the motorist and the police car separately. Let's break it down step by step:
a) To find out how many seconds pass before the police car reaches the motorist, we need to equate the distances traveled by both vehicles. At time t = 0, the motorist passes the police car, so the initial distance between them is zero.
Let's denote the time it takes for the police car to catch up with the motorist as t (in seconds). The distance traveled by the motorist during this time is given by:
Distance_motorist = Speed_motorist x Time = 15 m/s x t = 15t
The police car is initially at rest, so its initial speed is 0 m/s. It undergoes constant acceleration, so we can use the equation of motion:
Distance_police_car = Initial_velocity x Time + (1/2) x Acceleration x (Time)^2
Since the initial velocity of the police car is 0 m/s, the equation simplifies to:
Distance_police_car = (1/2) x Acceleration x (Time)^2 = (1/2) x 2 m/s^2 x t^2 = t^2
Since the two distances are the same (the police car catches up with the motorist), we can equate them:
15t = t^2
This is a quadratic equation. Rearranging it, we get:
t^2 - 15t = 0
Factoring out t, we have:
t(t - 15) = 0
This equation has two solutions: t = 0 (which we disregard since it represents the initial time when the motorist passes the police car) and t = 15. Therefore, it will take the police car 15 seconds to catch up with the motorist.
b) To find out how fast the police car is moving when it reaches the motorist, we need to calculate its final velocity at time t = 15 seconds. We can use the equation of motion:
Final_velocity_police_car = Initial_velocity + Acceleration x Time
Since the initial velocity is 0 m/s and the acceleration is 2 m/s^2, we substitute these values:
Final_velocity_police_car = 0 m/s + 2 m/s^2 x 15 s = 30 m/s
Therefore, the police car will be moving at a speed of 30 m/s when it reaches the motorist.
c) Finally, to determine the distance the police car has traveled when it reaches the motorist, we can substitute the value of t = 15 seconds into the equation for the distance traveled by the police car:
Distance_police_car = (1/2) x Acceleration x (Time)^2 = (1/2) x 2 m/s^2 x (15 s)^2 = (1/2) x 2 m/s^2 x 225 s^2 = 225 m
Hence, the police car will have traveled a total distance of 225 meters when it catches up with the motorist.
To solve this problem, we need to use the equations of motion and the concept of relative velocity.
a) To find the time it takes for the police car to catch up with the speeding motorist, we can use the equation:
Distance = Speed × Time
The distance the motorist travels in time t is given by:
Distance = Speed × Time = 15 m/s × t
The distance the police car travels in time t is given by:
Distance = Initial Speed × Time + (1/2) × Acceleration × Time²
Initially, the police car is at rest and then it starts accelerating at a rate of 2 m/s². So, the equation becomes:
Distance = 0 × t + (1/2) × 2 m/s² × t² = t²
To find the time when the police car catches up with the motorist, we need to set these two distances equal to each other:
15t = t²
Dividing both sides by t gives:
15 = t
Therefore, it takes 15 seconds before the police car reaches the motorist.
b) To find the speed of the police car when it reaches the motorist, we can use the equation:
Final Speed = Initial Speed + Acceleration × Time
The initial speed of the police car is 0 m/s, and the time it takes to reach the motorist is 15 seconds. The acceleration is given as 2 m/s². Substituting these values into the equation:
Final Speed = 0 + 2 m/s² × 15 s = 30 m/s
Therefore, the police car is moving at a speed of 30 m/s when it reaches the motorist.
c) To find the distance the police car has traveled when it reaches the motorist, we can use the equation:
Distance = Initial Speed × Time + (1/2) × Acceleration × Time²
The initial speed of the police car is 0 m/s, and the time it takes to reach the motorist is 15 seconds. The acceleration is given as 2 m/s². Substituting these values into the equation:
Distance = 0 × 15 s + (1/2) × 2 m/s² × (15 s)² = 225 m
Therefore, the police car has traveled a distance of 225 meters when it reaches the motorist.