A sprinter, running a 100-meter race, can accelerate with a constant acceleration for 3.50 s before reaching his top speed. He can then can maintain that speed for the rest of the race. He runs the 100 meters in 10.0 s.
let x = top speed
(3.5 * x/2) + 6.5 x = 100 m ... 8.25 x = 100 m
To find the sprinter's acceleration and top speed, we can use the kinematic equations of motion. Let's break it down step by step:
Step 1: Find the acceleration (a) during the first 3.50 s of the race.
We need to find the acceleration during this initial part of the race. Since the acceleration is constant, we can use the equation:
a = (vf - vi) / t
where
a = acceleration
vf = final velocity (top speed)
vi = initial velocity (0 for a stationary start)
t = time taken to reach top speed (3.50 s)
In this case, we want to find the acceleration, so let's rearrange the equation:
a = (vf - vi) / t
a = (vf - 0) / 3.50 s
a = vf / 3.50 s
Step 2: Find the top speed (vf).
We know that the sprinter reaches his top speed after running for 3.50 s. During this time, he accelerates with constant acceleration (a). We can use the equation of motion:
vf = vi + a * t
where
a = acceleration
vi = initial velocity (0 for a stationary start)
t = time taken to reach top speed (3.50 s)
Plugging in the given values:
vf = 0 + a * 3.50 s
vf = a * 3.50 s
Step 3: Find the distance covered during acceleration phase.
The distance covered during the acceleration phase is given by the equation:
distance = (vi + vf) / 2 * t
where
vi = initial velocity (0 for a stationary start)
vf = final velocity (top speed)
t = time taken to reach top speed (3.50 s)
Plugging in the given values:
distance = (0 + vf) / 2 * t
distance = vf / 2 * t
Step 4: Find the remaining distance covered after reaching top speed.
The remaining distance is given by subtracting the distance covered during the acceleration phase from the total race distance.
remaining distance = total distance - distance covered during acceleration phase
remaining distance = 100 m - distance
Step 5: Find the time taken for the remaining distance.
The time taken for the remaining distance is given by:
time = remaining distance / top speed
Step 6: Calculate the total race time.
The total race time is the sum of the time taken during the acceleration phase and the time taken for the remaining distance.
total race time = time taken during acceleration phase + time taken for remaining distance
Now, let's substitute the equations and calculate the values step-by-step.
To determine the sprinter's top speed, we can use the formulas for accelerated motion.
The formula to calculate velocity during an accelerated motion is:
v = u + at
Where,
v is the final velocity,
u is the initial velocity,
a is the acceleration, and
t is the time.
In this case, the sprinter starts from rest and accelerates with a constant acceleration for 3.50 seconds before reaching his top speed.
Since the sprinter starts from rest, the initial velocity (u) is 0 m/s.
The final velocity (v) at the end of the acceleration phase will be the sprinter's top speed.
The time (t) for the acceleration phase is given as 3.50 seconds.
So, the formula becomes:
v = 0 + a * 3.50
To calculate the acceleration (a), we need additional information.
V = Vo + a*T = 0 + a*3.5 = 3.5a m/s. = velocity reached during acceleration
d1 = 0.5(3.5a*3.5) = 6.125a meters = distance covered during acceleration
d2 = V*T = 3.5a * (10-3.5) = 22.75a meters = dist. covered during last 6.5 s
of race.
d1 + d2 = 100 m.
6.125a + 22.75a = 100
a = 3.5 m/s^2.