A ball starts from rest and rolls down a hill with uniform acceleration, traveling 150 m during the second 5.0 s of its motion. How far did it roll during the first 5.0 s of motion?
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To find how far the ball rolled during the first 5.0 seconds of motion, we need to use the equations of motion.
Let's denote the initial velocity as u, time as t, acceleration as a, and distance as s.
First, we can determine the acceleration of the ball by using the given information that it rolls down the hill with uniform acceleration. Since the ball starts from rest and travels 150 m during the second 5.0 s, we know that the average velocity of the ball during this time is (150 m) / (5.0 s) = 30 m/s.
Since the ball starts from rest, the initial velocity is u = 0 m/s, and we can calculate the acceleration using the formula:
average velocity = (initial velocity + final velocity) / 2
30 m/s = (0 m/s + final velocity) / 2
Solving for the final velocity, we get:
final velocity = 2 * (30 m/s) = 60 m/s
Now that we have the final velocity and the time for the second 5.0 seconds of motion (t = 5.0 s), we can calculate the acceleration using the equation:
final velocity = initial velocity + acceleration * time
60 m/s = 0 m/s + acceleration * 5.0 s
Solving for the acceleration, we get:
acceleration = 60 m/s / 5.0 s = 12 m/s^2
Now that we have the acceleration, we can find the distance the ball rolled during the first 5.0 seconds using the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since the initial velocity is zero, the equation simplifies to:
distance = (1/2) * acceleration * time^2
Plugging in the values, we get:
distance = (1/2) * (12 m/s^2) * (5.0 s)^2
distance = (1/2) * 12 m/s^2 * 25 s^2
distance = 6 m/s^2 * 25 s^2
distance = 150 m
Therefore, the ball rolled a distance of 150 meters during the first 5.0 seconds of its motion.