on a train traveling at a velocity of 45 miles per hour, ally slows the train down to a stop over a distance of 600 meters. what is the acceleration? (answer in m/s squared)
To calculate the acceleration, we need to use the equation for acceleration:
acceleration (a) = change in velocity (Δv) / time taken (Δt)
First, let's convert the velocity from miles per hour to meters per second. We know that 1 mile is equal to 1609.34 meters and 1 hour is equal to 3600 seconds. So, we have:
45 miles per hour = (45 miles/hour) * (1609.34 meters/mile) * (1 hour/3600 seconds) = 20.11 meters per second (approximately)
Now, we need to find the change in velocity (Δv). Since the train slows down to a stop, its final velocity (vf) is 0 m/s, and its initial velocity (vi) is 20.11 m/s. Thus, Δv = vf - vi = 0 - 20.11 = -20.11 m/s.
Next, we need to find the time taken (Δt). We are given the distance (600 meters), but not the time. However, we can use another formula to calculate the time:
distance (d) = initial velocity (vi) * time (Δt) + (1/2) * acceleration (a) * time^2
In this case, the final velocity (vf) is 0, since the train comes to a stop. Plugging in the values we know, we have:
600 meters = (20.11 m/s) * Δt + (1/2) * a * Δt^2
Now, we need to solve this equation for the time (Δt). Since we have a quadratic equation, we will use the quadratic formula:
Δt = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, let's consider:
a = (1/2) * acceleration (a)
b = initial velocity (vi)
c = -600 meters
Plugging in these values, we have:
Δt = (-vi ± √(vi^2 - 4 * (1/2) * a * -600)) / (2 * (1/2) * a)
We can simplify this equation and solve for Δt.
Once we have the time taken (Δt), we can finally calculate the acceleration (a) using the formula:
acceleration (a) = Δv / Δt
Plug in the values for Δv and Δt we found earlier and calculate the result to get the acceleration in meters per second squared (m/s²).