A pilot begins a race at a speed of 755.0 km/h and accelerates at a constant uniform rate for 63.21 s. The pilot cross the finish line with a speed of 777.0 km/h. From this data, calculate the length of the course.
Well, let's break it down! The pilot starts with a speed of 755.0 km/h and accelerates for 63.21 seconds until reaching a speed of 777.0 km/h. Now, to calculate the length of the course, we need to find the average speed.
But before we calculate anything, let me tell you a joke: Why don't scientists trust atoms? Because they make up everything!
Now, back to the calculation. To find the average speed, we'll use the formula:
Average speed = (initial speed + final speed) / 2
So, the average speed is (755.0 + 777.0) / 2 = 766.0 km/h.
Now, using the time taken (63.21 seconds) and the average speed (766.0 km/h), we can calculate the length of the course using the formula:
Length = Speed × Time
However, we need to convert the time to hours instead of seconds:
63.21 seconds = 63.21 / 3600 hours ≈ 0.0176 hours
Now we can calculate the length:
Length = 766.0 km/h × 0.0176 hours
And the answer is... drum roll, please... approximately 13.48 kilometers!
So, the length of the course is around 13.48 kilometers. Just like the length of a good dad joke!
To solve this problem, we need to use the kinematic equation:
v = u + at
where:
v = final speed
u = initial speed
a = acceleration
t = time
We also need to convert the speeds from km/h to m/s since the SI unit for acceleration is in m/s^2.
Given:
u = 755.0 km/h
v = 777.0 km/h
t = 63.21 s
Converting the speeds to m/s:
u = (755.0 km/h) * (1000 m/1 km) * (1 h/3600 s) = 209.72 m/s
v = (777.0 km/h) * (1000 m/1 km) * (1 h/3600 s) = 216.25 m/s
Since the acceleration is constant, we can use the average speed formula:
v_avg = (u + v) / 2
Substituting the given values:
v_avg = (209.72 m/s + 216.25 m/s) / 2 = 212.985 m/s
Now, let's solve for the acceleration (a):
a = (v - u) / t
Substituting the given values:
a = (216.25 m/s - 209.72 m/s) / 63.21 s = 6.53 m/s^2
Now, we can use the equations of motion:
v^2 = u^2 + 2as
Solving for s (the length of the course):
s = (v^2 - u^2) / (2a)
Substituting the given values:
s = (216.25 m/s)^2 - (209.72 m/s)^2 / (2 * 6.53 m/s^2) = 233.42 m
Therefore, the length of the course is approximately 233.42 meters.
To calculate the length of the course, we need to use the equation of motion that relates speed, acceleration, and time.
The equation is:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
From the given information, we have:
Initial velocity (u) = 755.0 km/h
Final velocity (v) = 777.0 km/h
Time (t) = 63.21 s
Now let's calculate the acceleration (a) using the equation:
a = (v - u) / t
Substituting the given values:
a = (777.0 km/h - 755.0 km/h) / 63.21 s
To calculate the acceleration, we need to convert the velocities from km/h to m/s because the unit of time is in seconds.
1 km/h = 0.2778 m/s
Substituting the velocities in m/s:
a = ((777.0 km/h - 755.0 km/h) * 0.2778 m/s) / 63.21 s
Now calculate the acceleration:
a = (22.0 km/h * 0.2778 m/s) / 63.21 s
a ≈ 0.0969 m/s²
Now we have the acceleration value. To find the length of the course, we need to calculate the distance traveled during the acceleration phase.
We can use the following equation to calculate the distance (S) during uniform acceleration:
S = ut + (1/2)at²
Where:
S = Distance
u = Initial velocity
t = Time
a = Acceleration
Substituting the given values:
S = (755.0 km/h * 0.2778 m/s * 63.21 s) + (1/2 * 0.0969 m/s² * (63.21 s)²
Calculating the distance traveled during the acceleration phase:
S = (755.0 km/h * 0.2778 m/s * 63.21 s) + (1/2 * 0.0969 m/s² * (63.21 s)²
Finally, calculate the total distance of the course by considering the distance traveled during the acceleration phase and adding it to the distance covered at the finish line (assuming the pilot maintains a constant speed after acceleration).