After 35 minutes of running, at the 8-{\rm km} point in a 9-{\rm km} race, you find yourself 200m behind the leader and moving at the same speed.
What should your acceleration be if you are to catch up by the finish line? Assume that the leader maintains a constant speed throughout the entire race.
To calculate the acceleration needed to catch up to the leader by the finish line, we need to determine the time it would take to cover the remaining 1 km distance and the speed at which the leader is moving.
First, let's find out the speed at which you are running at the 8 km point. Since you have been running for 35 minutes, which is equal to 35/60 = 0.5833 hours, we can calculate your speed using the formula:
Speed = Distance / Time
Speed = 8 km / 0.5833 hours
Speed = 13.726 km/h
Now, let's calculate the speed at which the leader is moving. Since you are 200 m behind the leader at the 8 km point, the leader has already covered an additional 200 m (or 0.2 km). So, the distance covered by the leader is 8 km + 0.2 km = 8.2 km.
To find the speed of the leader, we can use the formula:
Speed = Distance / Time
Now, we know that the time is 35 minutes, so we need to convert it to hours:
Time = 35/60 = 0.5833 hours
Speed = 8.2 km / 0.5833 hours
Speed = 14.081 km/h
Now that we have determined your speed and the leader's speed, we can find the acceleration needed to catch up.
Acceleration is the change in velocity over time. In this case, the change in velocity is the difference between your speed and the leader's speed.
Change in velocity = Leader's speed - Your speed
Change in velocity = 14.081 km/h - 13.726 km/h
Change in velocity = 0.355 km/h
Since we want to determine the acceleration in meters per second squared (m/s^2), we need to convert the velocity from kilometers per hour (km/h) to meters per second (m/s):
Conversion factor: 1 km/h = 1000 m/3600 s
Change in velocity = 0.355 km/h * (1000 m/3600 s)
Change in velocity = 0.0986 m/s
Now, we need to find the time it would take to cover the remaining 1 km distance. Since your speed is the same as the leader's speed, we can use the formula:
Time = Distance / Speed
Time = 1 km / 13.726 km/h
Time = 0.0729 hours
Finally, we can calculate the acceleration using the formula:
Acceleration = Change in velocity / Time
Acceleration = 0.0986 m/s / 0.0729 hours
Acceleration = 1.352 m/s^2
So, in order to catch up to the leader by the finish line, your acceleration should be 1.352 m/s^2.