An object glides on a horizontal tabletop with a coefficient of kinetic friction of 0.65. If it comes to rest after 0.9 seconds, what was its initial velocity?
See my reply to your earlier question.
To find the initial velocity of the object, we can use the formula for motion under the influence of friction:
\[ v_{\text{f}} = v_{\text{0}} + at \]
In this formula, \( v_{\text{f}} \) represents the final velocity, \( v_{\text{0}} \) represents the initial velocity, \( a \) represents the acceleration, and \( t \) represents the time taken.
In this scenario, the object comes to rest after 0.9 seconds, so we have:
\[ v_{\text{f}} = 0 \]
\[ t = 0.9 \]
We also need to determine the acceleration. The acceleration \( a \) in this case is the product of the coefficient of kinetic friction (\( \mu \)) and the acceleration due to gravity (\( g \)):
\[ a = \mu \cdot g \]
Given that the coefficient of kinetic friction is 0.65, we can substitute this value into the equation. The acceleration due to gravity (\( g \)) is approximately 9.8 m/s\(^2\).
\[ a = 0.65 \cdot 9.8 \]
Now, we can use these values to solve for the initial velocity (\( v_{\text{0}} \)).
\[ v_{\text{f}} = v_{\text{0}} + at \]
\[ 0 = v_{\text{0}} + a \cdot 0.9 \]
Substituting the values of \( a \) and \( t \) into the equation:
\[ 0 = v_{\text{0}} + (0.65 \cdot 9.8) \cdot 0.9 \]
Simplifying further:
\[ -v_{\text{0}} = 0.65 \cdot 9.8 \cdot 0.9 \]
Finally, we can solve for \( v_{\text{0}} \) by multiplying the right side of the equation:
\[ v_{\text{0}} = -0.65 \cdot 9.8 \cdot 0.9 \]
Evaluating this expression gives us the initial velocity of the object.