A ball starts from rest and rolls down a hill with uniform acceleration, traveling 170 m during the second 5.5 s of its motion
How far did it roll during the first 5.5 s of motion?
x=
To find the distance the ball rolled during the first 5.5 s of motion, we need to use the formula for distance traveled with uniform acceleration. This formula is:
x = ut + (1/2)at^2
Where:
x = distance traveled
u = initial velocity (in this case, the ball starts from rest, so u = 0)
t = time
a = acceleration
Since the ball starts from rest, its initial velocity is 0, so the first term in the equation becomes 0. Therefore, we can simplify the equation to:
x = (1/2)at^2
Next, we need to find the acceleration of the ball. The problem states that the ball has uniform acceleration. We are given the distance traveled (170 m) during the second 5.5 s, so we can use this information to determine the acceleration.
We can use the formula for average acceleration:
a = (v - u) / t
Where:
a = acceleration
v = final velocity
u = initial velocity
t = time
Since the initial velocity is 0 (ball starts from rest), the equation becomes:
a = v / t
We are given that the ball traveled 170 m during the second 5.5 s of motion. So, during this time, the final velocity (v) would be the velocity at the end of this time interval.
Now, we know that velocity is the rate of change of distance with time. Therefore, we can write:
v = (change in distance) / (change in time)
v = 170 m / 5.5 s
Now, we need to plug this value into the equation for acceleration:
a = v / t
a = (170 m / 5.5 s) / 5.5 s
Evaluate the expression to find the acceleration:
a = 6.909 m/s^2
Now that we know the acceleration, we can plug this value into the formula for distance traveled during the first 5.5 s:
x = (1/2)at^2
x = (1/2)(6.909 m/s^2)(5.5 s)^2
Evaluate the expression to find the distance traveled during the first 5.5 s of motion:
x = 81.055 m
Therefore, the ball rolled 81.055 m during the first 5.5 s of motion.