A sprinter can accelerate with constant acceleration for 2.70 s before reaching top speed. He can run the 100-meter dash in 10 s. What is his speed as he crosses the finish line?

Please show me all the steps to solving this problem.

Thanks

Ok, during the acceleration phase, his average velocity was Vf/2, during the rest of the race, the velocity is Vf.

distance= distance accelerating+ steady running distance
100 = Vf/2 * 2.70+ Vf*(10-2.70)
100= Vf(10-2.70/2)
solve for Vf

To solve this problem, we need to use the equation of motion:

v = u + at

where:
v = final velocity (speed) of the sprinter
u = initial velocity (speed) of the sprinter
a = acceleration of the sprinter
t = time interval

First, let's calculate the acceleration of the sprinter. We know that the sprinter accelerates with constant acceleration for 2.70 seconds before reaching top speed. Therefore, we can use the equation:

a = (v - u) / t

Substituting the given values, we have:

a = (v - 0) / 2.70

Since the sprinter is already moving when he starts accelerating, his initial velocity u is zero. Thus, the equation simplifies to:

a = v / 2.70

Now, let's find the acceleration value. We can use the information given in the problem to do this. The sprinter can run the 100-meter dash in 10 seconds. Since the distance traveled during acceleration is not given, we'll consider the entire race distance of 100 meters as the distance covered during acceleration. We can use the equation of motion:

s = ut + (1/2)at^2

where:
s = distance
u = initial velocity (speed)
t = time interval
a = acceleration

The distance traveled during acceleration can be represented by s. Initially, the sprinter is at rest (u = 0), and the time interval is given as 2.70 seconds. Thus, the equation simplifies to:

100 = 0 + (1/2)(a)(2.70)^2

Now, we can solve for the acceleration a:

100 = (1/2)(a)(2.70)^2

100 = (1/2)(a)(7.29)

200 = 7.29a

a = 200 / 7.29

a ≈ 27.43 m/s^2

Now that we have found the acceleration, we can substitute it back into the equation from earlier:

a = v / 2.70

27.43 m/s^2 = v / 2.70

v = 27.43 m/s^2 * 2.70

v ≈ 74.115 m/s

Therefore, the sprinter's speed as he crosses the finish line is approximately 74.115 m/s.

To find the sprinter's speed as he crosses the finish line, we need to calculate his acceleration. Here are the step-by-step calculations:

Step 1: Convert the time taken to reach top speed to acceleration.
Acceleration (a) = change in velocity (Δv) / time taken (Δt) = (0 - initial velocity) / time taken
Since the sprinter starts from rest, the initial velocity is 0 m/s.
Thus, the acceleration is a = 0 - 0 / 2.70 s = 0 m/s².

Step 2: Calculate the distance covered during the acceleration phase.
During the 2.70 s of constant acceleration, the distance covered can be calculated using the equation:
Distance (d) = initial velocity * time + (1/2) * acceleration * time²
Since the initial velocity is 0 m/s and the time is 2.70 s, the distance can be calculated as:
d = 0 * 2.70 + (1/2) * 0 * (2.70)² = 0 m.

Step 3: Calculate the distance covered after reaching top speed.
The remaining distance for the sprinter to cross the finish line is 100 m (as given).
Therefore, the distance covered after reaching top speed is 100 m - 0 m = 100 m.

Step 4: Calculate the time taken to cover the remaining distance.
The time taken to cover the remaining distance can be calculated using the equation:
Time (t) = Distance (d) / Speed (v)
Since we want to find the speed, we rearrange the equation as:
Speed (v) = Distance (d) / Time (t)
Plugging in the values, we get:
Speed (v) = 100 m / 10 s = 10 m/s.

Therefore, the sprinter's speed as he crosses the finish line is 10 m/s.