A bicyclist starts a timed race at 6 mph In order to win, he must average 21 mph Assuming constant acceleration from the start, how fast must he be traveling at the end of the race?

If the average must be 21,

(V1 + V2)/2 = 21

V1 + V2 = 42

V2 = 42 - V1 = 42 - 6 = ___ mph

V2 is the final velocity required
V1 is the initial velocity

To find out how fast the bicyclist must be traveling at the end of the race, we can use the formula for average speed:

Average speed = Total distance / Total time

We know that the average speed the bicyclist must achieve is 21 mph. Let's assume the total distance of the race is "d" miles.

To calculate the total time, we need to consider the time it takes for the bicyclist to accelerate to the final speed, as well as the time spent at that final speed.

Let's first calculate the time required for acceleration. We can use the equation:

v = u + at

where:
v = final velocity
u = initial velocity
a = acceleration
t = time

Here, the bicyclist starts with an initial velocity of 6 mph and must accelerate to the final velocity. Let's assume the acceleration is "a" mph^2. The time required for acceleration can be expressed as:

t = (v - u) / a

The acceleration time is given by (t_acc):

t_acc = (v - 6) / a

Now, let's determine the time spent at the final velocity. We can calculate this using the formula:

t_const = (d - vt_acc) / v

Here, vt_acc is the distance traveled during the acceleration period, which is given by:

vt_acc = (1/2) * a * t_acc^2

Finally, we can calculate the total time for the race (t_total) as:

t_total = t_acc + t_const

We can now substitute the values into the equation for average speed:

21 = d / t_total

Let's rearrange this equation to solve for "d":

d = 21 * t_total

Now, we will substitute the values of t_acc and t_total into this equation and solve for "d".

To determine the speed the bicyclist must be traveling at the end of the race, we need to find the average speed for the entire race. The average speed is given by the formula: average speed = total distance / total time.

Let's assume the race lasts for a certain period of time, which we'll denote as "t".

Now, we can find the total distance covered by the bicyclist during the race. Since the average speed is 21 mph, we can use the formula for average speed: average speed = total distance / t. Rearranging the formula, we get: total distance = average speed * t.

Since the bicyclist starts at a speed of 6 mph, we can calculate the time it takes to reach the average speed of 21 mph. The time taken to accelerate to the average speed is denoted as "a". We can calculate this time using the formula: a = (final speed - initial speed) / acceleration. In this case, the initial speed is 6 mph, the final speed is 21 mph, and we'll assume constant acceleration.

Now, we can calculate the time taken to accelerate to the average speed and the time remaining in the race. The total time for the race, denoted as "T", is equal to the time to accelerate to the average speed plus the remaining time: T = a + (t - a).

Finally, we can substitute the values back into the formula for average speed to find the total distance covered during the race: average speed = total distance / T.

Let's solve the problem step by step:

Step 1: Calculate the time taken to accelerate to the average speed.
initial speed = 6 mph
final speed = 21 mph
acceleration = ?
a = (final speed - initial speed) / acceleration
a = (21 mph - 6 mph) / acceleration

Step 2: Calculate the remaining time in the race.
t - a = remaining time

Step 3: Calculate the total time for the race.
T = a + (t - a)

Step 4: Calculate the total distance covered during the race.
average speed = total distance / T

By following these steps and solving the equations, we can determine the speed the bicyclist must be traveling at the end of the race.