A speedboat uniformly increases its velocity from 25 m/s to the west to 35 m/s to the west. How long does it take the boat to travel 250 , west while undergoing the acceleration?
C=43
To find the time it takes for the boat to travel a certain distance while undergoing acceleration, we can use the formula:
\[ t = \frac{v_f - v_i}{a} \]
Where:
- \( t \) is the time taken
- \( v_f \) is the final velocity
- \( v_i \) is the initial velocity
- \( a \) is the acceleration
In this case, the boat is traveling to the west, so the velocities will be negative. The initial velocity, \( v_i \), is -25 m/s, and the final velocity, \( v_f \), is -35 m/s. The change in velocity, \( \Delta v \), is:
\[ \Delta v = v_f - v_i = -35 \, \text{m/s} - (-25 \, \text{m/s}) = -10 \, \text{m/s} \]
We are given the distance, which is 250 m, and the velocity is changing uniformly, so we can find the acceleration, \( a \), using the equation:
\[ a = \frac{\Delta v}{t} \]
We can rearrange this equation to solve for time:
\[ t = \frac{\Delta v}{a} = \frac{(-10 \, \text{m/s})}{a} \]
Now we have all the required values to find the time taken. However, we need to find the acceleration first.
Since we are not given the acceleration directly, we can use a kinematic equation to solve for it. The kinematic equation we can use is:
\[ v_f^2 = v_i^2 + 2ad \]
Where:
- \( v_f \) is the final velocity
- \( v_i \) is the initial velocity
- \( a \) is the acceleration
- \( d \) is the distance traveled
Rearranging the equation to solve for acceleration:
\[ a = \frac{v_f^2 - v_i^2}{2d} \]
Substituting the given values:
\[ a = \frac{(-35 \, \text{m/s})^2 - (-25 \, \text{m/s})^2}{2 \times 250 \, \text{m}} \]
Calculating the numerator:
\[ (-35 \, \text{m/s})^2 - (-25 \, \text{m/s})^2 = 1225 \, \text{m/s}^2 - 625 \, \text{m/s}^2 = 600 \, \text{m/s}^2 \]
Now substituting this value into the equation for acceleration:
\[ a = \frac{600 \, \text{m/s}^2}{2 \times 250 \, \text{m}} \]
Simplifying:
\[ a = \frac{600 \, \text{m/s}^2}{500 \, \text{m}} = 1.2 \, \text{m/s}^2 \]
Now we can substitute the calculated acceleration into the equation for time:
\[ t = \frac{(-10 \, \text{m/s})}{1.2 \, \text{m/s}^2} \]
Calculating:
\[ t = -8.33 \, \text{s} \]
Since time cannot be negative, we take the magnitude of the result:
\[ |t| = 8.33 \, \text{s} \]
Therefore, it takes the speedboat approximately 8.33 seconds to travel 250 meters to the west while undergoing the given acceleration.