An athlete wants to complete the 4000 m run in 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 to 4.0 m/s^2, how much longer will it take her to complete the race?

She is traveling at 3200m/12(60)sec

figure out that.

d=vi*t+1/2 (4)t^2
d is 800, vi is above, solve for t.

To determine the time it will take for the athlete to complete the race, we need to calculate the time it takes for her to cover the remaining distance after 10.0 minutes and the time it takes for her to accelerate to the final speed of 4.0 m/s.

Step 1: Calculate the remaining distance after 10 minutes of running:
Distance covered in 10 minutes = speed × time
Distance covered = (4000 m / 12.0 min) × 10.0 min = 3333.33 m

Remaining distance = Total distance - Distance covered
Remaining distance = 4000 m - 3333.33 m = 666.67 m

Step 2: Calculate the time it will take to cover the remaining distance at the final speed:
Using the equation for uniformly accelerated motion:
Distance = Initial velocity × time + (0.5 × acceleration × time^2)

Since the initial velocity is unknown, we can assume it to be zero.
Distance = 0 × time + (0.5 × acceleration × time^2)
666.67 m = 0.5 × (4.0 m/s^2) × time^2

Solving for time:
2 × 666.67 m/(4.0 m/s^2) = time^2
3333.34 s/(4.0 m/s^2) = time^2
833.34 s = time^2
time = √833.34 s ≈ 28.87 s

Therefore, it will take the athlete approximately 28.87 seconds longer to complete the race at an accelerated speed.

To determine how much longer it will take the athlete to complete the race, we need to find out the time it takes for her to cover the remaining distance of 800 m with an acceleration of 4.0 m/s^2.

First, let's calculate the initial velocity (v_0) of the athlete after 10.0 minutes (600 seconds) of running at a constant speed.
Using the formula v = d/t, where v is the velocity, d is the distance, and t is the time, we can rearrange the formula to solve for v.
Since the velocity remains constant, we can assume that the initial velocity (v_0) is equal to the average velocity of the athlete during the initial 10.0 minutes.
v_0 = d/t = 800 m / 600 s = 1.33 m/s

Now, we can use the kinematic equation to find the time it takes the athlete to cover the remaining distance (800 m) with an acceleration of 4.0 m/s^2.
The kinematic equation that relates initial velocity (v_0), final velocity (v_f), acceleration (a), and time (t) is:
v_f = v_0 + at
where v_f is the final velocity, a is the acceleration, and t is the time.

Here, the initial velocity (v_0) is 1.33 m/s, the acceleration (a) is 4.0 m/s^2, and the final velocity (v_f) is unknown. The distance (d) is 800 m.
Since the athlete wants to cover the remaining distance (800 m), the final velocity (v_f) would be the velocity at the end of the race, which we can assume is zero.

Plugging in the values into the formula:
0 = 1.33 m/s + (4.0 m/s^2) * t

Rearranging the equation to solve for time (t):
4.0 m/s^2 * t = -1.33 m/s
t = (-1.33 m/s) / (4.0 m/s^2) = -0.3325 s

Since time cannot be negative, we can conclude that the athlete will take an additional 0.3325 seconds to complete the race.