To determine the acceleration of the ball as it is pushed off the cliff, we can use one of the fundamental equations of motion. The equation that relates acceleration ([tex]\( a \)[/tex]), initial velocity ([tex]\( v_i \)[/tex]), final velocity ([tex]\( v_f \)[/tex]), and time ([tex]\( t \)[/tex]) is:
[tex]\[ a = \frac{v_f - v_i}{t} \][/tex]
Let's break this down step-by-step:
1. Identify known values:
- Initial velocity ([tex]\( v_i \)[/tex]): The ball starts from rest, so [tex]\( v_i = 0 \, \text{m/s} \)[/tex].
- Final velocity ([tex]\( v_f \)[/tex]): The ball reaches [tex]\( 18 \, \text{m/s} \)[/tex] downward.
- Time ([tex]\( t \)[/tex]): The time taken to reach this velocity is [tex]\( 6.4 \)[/tex] seconds.
2. Substitute the known values into the equation:
[tex]\[ a = \frac{18 \, \text{m/s} - 0 \, \text{m/s}}{6.4 \, \text{s}} \][/tex]
3. Perform the calculation:
[tex]\[ a = \frac{18 \, \text{m/s}}{6.4 \, \text{s}} \][/tex]
4. Simplify the fraction:
[tex]\[ a = 2.8125 \, \text{m/s}^2 \][/tex]
Hence, the acceleration of the ball during this time frame is [tex]\( 2.8125 \, \text{m/s}^2 \)[/tex].
