A race car accelerates from 50 m/s to 75 m/s over a distance of 400 m. What is the race car’s acceleration?
To find the race car's acceleration, we can use the acceleration formula:
Acceleration (a) = (final velocity (v) - initial velocity (u)) / time taken (t)
Given:
Initial velocity (u) = 50 m/s
Final velocity (v) = 75 m/s
First, we need to find the time taken (t). We can use the equation:
Distance (d) = (initial velocity (u) + final velocity (v)) / 2 * time taken (t)
Where:
Distance (d) = 400 m
Rearranging the equation to solve for time taken (t):
2 * time taken (t) = Distance (d) / (initial velocity (u) + final velocity (v))
time taken (t) = (Distance (d) / (initial velocity (u) + final velocity (v))) / 2
Substituting the given values into the equation:
time taken (t) = (400 m) / ((50 m/s + 75 m/s) / 2) / 2
Simplifying the equation:
time taken (t) = (400 m) / (125 m/s) / 2
time taken (t) = (400 m) / 125 m/s / 2
time taken (t) = (400 m) / 125 m/s / 2
time taken (t) = 1.6 s
Now, we can substitute the values back into the acceleration formula:
Acceleration (a) = (final velocity (v) - initial velocity (u)) / time taken (t)
Acceleration (a) = (75 m/s - 50 m/s) / 1.6 s
Acceleration (a) = 25 m/s / 1.6 s
Acceleration (a) = 15.625 m/s^2
Therefore, the race car's acceleration is 15.625 m/s^2.
To find the race car's acceleration, we can use the formula:
Acceleration = (Change in velocity) / (Change in time)
But since we are given the change in distance instead of time, we first need to find the change in time. To do that, we can use the formula:
Distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)
We know the initial velocity is 50 m/s, the final velocity is 75 m/s, and the distance is 400 m. Let's proceed to calculate the change in time.
Rearranging the formula for distance, we get:
400 = (50 * Time) + (0.5 * Acceleration * Time^2)
Rearranging further, we have a quadratic equation:
0.5 * Acceleration * Time^2 + 50 * Time - 400 = 0
We can solve this equation using the quadratic formula. Considering the positive root of the equation will give us the time it takes for the car to accelerate from 50 m/s to 75 m/s.
Once we know the change in time, we can substitute it into the acceleration formula:
Acceleration = (Change in velocity) / (Change in time)
Substituting the values, we get:
Acceleration = (75 - 50) / (change in time)
Now, let's calculate the change in time using the quadratic formula and find the race car's acceleration.