Question

kayla is measuring how long it takes a ball to roll down a hill. the ball has a starting velocity of latex: 2.6 m/s. the ball rolls down the hill for latex: 5.5 seconds, and when it reaches the bottom, it has a velocity of latex: 8.0 m/s. how far did the ball roll during this time frame?

Answer 1

To find the distance the ball rolled down the hill, you can use the equations of motion. In this case, you have the initial velocity (u), final velocity (v), and time (t). You can use the following equation:

\[s = ut + \frac{1}{2}at^2\]

Where:
- \(s\) is the distance traveled
- \(u\) is the initial velocity (2.6 m/s)
- \(t\) is the time (5.5 seconds)
- \(a\) is the acceleration (which we can calculate using the change in velocity)

First, calculate the acceleration (a):
\[a = \frac{v - u}{t}\]

Where:
- \(v\) is the final velocity (8.0 m/s)
- \(u\) is the initial velocity (2.6 m/s)
- \(t\) is the time (5.5 seconds)

\[a = \frac{8.0 m/s - 2.6 m/s}{5.5 s} = \frac{5.4 m/s}{5.5 s} \approx 0.982 m/s^2\]

Now that you have the acceleration, you can find the distance (s) the ball rolled down the hill:

\[s = (2.6 m/s) * (5.5 s) + \frac{1}{2} * (0.982 m/s^2) * (5.5 s)^2\]

Calculate each part separately:
\[s_1 = (2.6 m/s) * (5.5 s) = 14.3 m\]
\[s_2 = \frac{1}{2} * (0.982 m/s^2) * (5.5 s)^2 \approx 15.8 m\]

Now, add \(s_1\) and \(s_2\) to find the total distance:
\[s = s_1 + s_2 = 14.3 m + 15.8 m = 30.1 m\]

So, the ball rolled approximately 30.1 meters down the hill.