An Olympic athlete wants to complete the 4000-m run in less than 12.0 minutes. After exactly 10.0 minutes of running at a constant speed, she still has 800 m to go. If she then accelerates at a rate of 0.40 m/ s 2 , a) how much longer will it take her to complete the race?
Let's break down the information given:
- The athlete wants to complete the 4000-m run in less than 12.0 minutes.
- After 10.0 minutes of running at a constant speed, she still has 800m to go.
- She accelerates at a rate of 0.40 m/s^2.
To find out how much longer it will take her to complete the race, we need to calculate the time it takes for her to cover the remaining distance of 800m.
Using the equation for calculating distance with constant acceleration:
\[d = ut + \frac{1}{2} a t^2\]
where:
d = distance covered
u = initial speed
t = time taken
a = acceleration
We know that u = 0 m/s (since she started accelerating after 10.0 minutes), a = 0.40 m/s^2, and d = 800m. Let's rearrange the formula to solve for t:
\[d = \frac{1}{2} a t^2\]
\[800 = \frac{1}{2} \cdot 0.40 \cdot t^2\]
\[1600 = 0.20 \cdot t^2\]
\[t^2 = \frac{1600}{0.20}\]
\[t^2 = 8000\]
\[t = \sqrt{8000}\]
\[t \approx 89.44\]
So, it will take her approximately 89.44 seconds longer to complete the race.
b) will she achieve her desired time?
To find out how much longer it will take the athlete to complete the race, we need to calculate the time it takes for her to cover the remaining 800 m with an acceleration of 0.40 m/s^2.
We can use the equation of motion:
s = ut + (1/2)at^2
Where:
s = distance traveled
u = initial velocity
t = time taken
a = acceleration
We know that the initial velocity (u) is the speed at which she is running after 10.0 minutes, and the distance (s) is 800 m. The acceleration (a) is given as 0.40 m/s^2.
Let's calculate the time it will take for her to cover the remaining distance:
800 = (u * t) + (0.5 * 0.40 * t^2)
Simplifying the equation:
800 = (u * t) + (0.2 * t^2)
Since she is running at a constant speed after the initial 10.0 minutes, we can write u * t as just u.
800 = u + (0.2 * t^2)
Now, we need to find the value of u.
We know that distance = speed * time, so the initial velocity (u) can be calculated as:
u = (4000 - distance covered) / time taken
= (4000 - 3200) / 10
= 800 / 10
= 80 m/min
Now, we substitute the value of u into the equation:
800 = 80 + (0.2 * t^2)
Rearranging the equation:
0.2 * t^2 = 800 - 80
t^2 = 720 / 0.2
t^2 = 3600
t = √3600
t = 60 seconds
Therefore, it will take her an additional 60 seconds or 1 minute to complete the race.
To find out how much longer it will take the athlete to complete the race, we need to calculate the time it will take her to cover the remaining 800 meters after accelerating.
We know that the athlete has already run for 10 minutes and has 800 meters remaining. Since we want to find the time in seconds, we need to convert the 10 minutes to seconds. There are 60 seconds in a minute, so we can multiply 10 minutes by 60 seconds/minute to get 600 seconds.
Next, we need to calculate the time it will take for the athlete to cover the remaining 800 meters after accelerating. To do this, we can use the equation:
s = ut + (1/2)at^2
Where:
s = displacement (800 meters)
u = initial velocity (the velocity at the end of the 10 minutes)
a = acceleration (0.40 m/s^2)
t = time
Since the athlete has accelerated, her initial velocity is not zero. We know that the initial velocity is the rate of change of distance with time, which is the distance covered in 10 minutes divided by the time taken. So, we can calculate the initial velocity.
Initial velocity = (distance covered in 10 minutes) / (time taken)
Initial velocity = (4000 meters - 800 meters) / 600 seconds
Initial velocity = 3200 meters / 600 seconds
Initial velocity = 5.33 m/s
Now, we can substitute the values into the equation to solve for the time:
800 = (5.33)(t) + (1/2)(0.40)(t^2)
Rearranging the equation:
1/2(0.40)t^2 + 5.33t - 800 = 0
Now, we can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where:
a = 1/2(0.40) = 0.20
b = 5.33
c = -800
Plugging in the values:
t = (-(5.33) ± √((5.33)^2 - 4(0.20)(-800))) / (2(0.20))
Simplifying:
t = (-5.33 ± √(28.4089 + 640)) / 0.40
t = (-5.33 ± √(668.4089)) / 0.40
t = (-5.33 ± 25.865) / 0.40
Now, we can solve for the positive value of t:
t = (-5.33 + 25.865) / 0.4
t = 20.535 / 0.40
t ≈ 51.34 seconds
Therefore, it will take the athlete approximately an additional 51.34 seconds to complete the race after accelerating.