A police officer observes a motorist who is approaching at a uniform speed of 80km/h he in his patrol car starts chasing it as it just crosses him.after accelerating for 8 seconds at a constant rate he attains his top speed of 120km/h how long does it take to overtake the motorist and at what distance?
Just look at the solved problems linked below. This problem may be the most common in all of algebra and beginning physics :) Remember and hour is 3600 seconds.
To find out how long it takes for the police officer to overtake the motorist and at what distance, we need to calculate the relative acceleration and then use the kinematic equation.
Let's start by finding the relative acceleration of the police officer with respect to the motorist. The police officer's initial speed is 0 km/h, and after accelerating for 8 seconds, he attains his top speed of 120 km/h. Therefore, the officer's acceleration can be calculated as follows:
Acceleration (a) = (Final speed - Initial speed) / Time
= (120 km/h - 0 km/h) / 8 s
= 15 km/h/s
Now, let's calculate the time it takes for the police officer to catch up to the motorist. We can use the following kinematic equation:
Distance (d) = Initial speed * Time + (1/2) * Acceleration * Time^2
At the moment the police officer starts, the motorist has already crossed him. So, the initial distance between them can be assumed to be zero.
Setting the distance equation for both the motorist and the police officer equal to each other:
80 km/h * T + (1/2) * 0 * T^2 = 0 km/h * T + (1/2) * 15 km/h/s * T^2
Simplifying the equation:
40 T^2 = 30 T^2
Rearranging the equation:
10 T^2 = 0
This equation suggests that T = 0, which means that the police officer and the motorist meet at the same position. This implies that the police officer overtakes the motorist instantly.
Therefore, the time it takes for the police officer to overtake the motorist is 0 seconds, and they meet at the same position, i.e., zero distance.
To find the time it takes for the police officer to overtake the motorist and the distance covered, we can break down the problem step by step:
1. Convert the speeds from km/h to m/s to maintain consistent units:
- The motorist's speed is 80 km/h, so it is moving at (80 * 1000) / 3600 = 22.222 m/s.
- The police officer's top speed is 120 km/h, so he is moving at (120 * 1000) / 3600 = 33.333 m/s.
2. Calculate the time required for the police officer to reach his top speed:
- The police officer accelerated for 8 seconds to reach his top speed.
3. Determine the distance traveled during the acceleration phase:
- The distance covered during acceleration is given by the formula: distance = (initial velocity × time) + (0.5 × acceleration × time^2).
- Since the initial velocity is 0 m/s and the acceleration is unknown, we need to find the acceleration first.
4. Calculate the acceleration rate:
- The acceleration is the change in velocity (top speed - initial speed) divided by the time taken to accelerate.
- The acceleration rate is (33.333 m/s - 0 m/s) / 8 s = 4.167 m/s^2.
5. Calculate the distance covered during acceleration:
- distance = (0 × 8) + (0.5 × 4.167 × (8^2)) = 133.344 m.
6. Calculate the time it takes the police officer to overtake the motorist:
- The police officer needs to cover the distance the motorist has already traveled (since he starts chasing after the motorist passes him) and the distance covered during acceleration.
- The time taken to overtake is distance / relative speed.
- relative speed = top speed of the police officer - speed of the motorist = 33.333 m/s - 22.222 m/s = 11.111 m/s.
- time taken to overtake = (distance covered by motorist + distance during acceleration) / relative speed.
Now we can plug in the values and calculate the time it takes to overtake and the distance covered:
- Distance covered by the motorist = 22.222 m/s × time taken to overtake.
- Distance covered by the police officer during acceleration = 133.344 m.
Let's calculate the values step by step:
1. Calculate the time taken to overtake:
- time taken to overtake = (22.222 m/s × time taken to overtake) + 133.344 m / 11.111 m/s.
2. Simplify the equation by multiplying both sides by 11.111:
- 11.111 × time taken to overtake = 22.222 × time taken to overtake + 133.344.
3. Rearrange the equation to isolate the time taken to overtake:
- 22.222 × time taken to overtake - 11.111 × time taken to overtake = 133.344.
- 11.111 × time taken to overtake = 133.344.
- time taken to overtake = 133.344 / 11.111.
4. Calculate the time taken to overtake:
- time taken to overtake = 12 seconds.
5. Calculate the distance covered during acceleration:
- distance = 133.344 m.
Therefore, the police officer will overtake the motorist in 12 seconds and at a distance of 133.344 meters.