A race driver has made a pit stop to refuel. After refueling, he starts from rest and leaves the pit area with an acceleration whose magnitude is 5.8 m/s2; after 3.6 s he enters the main speedway. At the same instant, another car on the speedway and traveling at a constant velocity of 67.4 m/s overtakes and passes the entering car. The entering car maintains its acceleration. How much time is required for the entering car to catch up with the other car?
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To solve this problem, we need to determine how long it will take for the entering car to catch up with the other car on the speedway.
Let's break down the information given:
1. Acceleration of the entering car: 5.8 m/s².
2. Time taken by the entering car to enter the main speedway: 3.6 seconds.
3. Velocity of the other car on the speedway: 67.4 m/s.
First, we need to find out the distance covered by the entering car during the time it takes to enter the main speedway. We can use the kinematic equation:
\[d = ut + \frac{1}{2}at^2\]
where:
- d is the distance covered,
- u is the initial velocity (which is 0 since the car starts from rest),
- a is the acceleration, and
- t is the time taken.
Plugging in the values, we have:
\[d = 0 * 3.6 + \frac{1}{2} * 5.8 * (3.6)^2\]
Simplifying the equation, we get:
\[d = 0 + \frac{1}{2} * 5.8 * 12.96\]
\[d = 0 + 18.8544\]
\[d = 18.8544\,m\]
Now, we need to determine the time it will take for the entering car to catch up with the other car. Since they are traveling at different velocities, we can find the time needed using the equation:
\[t = \frac{d}{v}\]
where:
- t is the time taken,
- d is the distance covered by the entering car, and
- v is the relative velocity between the two cars.
In this case, the relative velocity will be the difference between the velocity of the other car on the speedway and the initial velocity of the entering car, which is 0:
\[v = 67.4\,m/s - 0\,m/s\]
\[v = 67.4\,m/s\]
Plugging in the values, we get:
\[t = \frac{18.8544\,m}{67.4\,m/s}\]
Simplifying the equation, we find:
\[t = 0.28\,s\]
Therefore, it will take approximately 0.28 seconds for the entering car to catch up with the other car on the speedway.