Answer :
To calculate the car's original speed before braking, we can use the equation of motion:
- \(v_f^2 = v_i^2 + 2a d\)
Where:
- \(v_f\) = final speed = 2.60 m/s
- \(v_i\) = initial speed (unknown)
- \(a\) = acceleration while braking (approximately constant)
- \(d\) = distance = 40.0 m
Rearranging the equation, we have:
- \(v_i^2 = v_f^2 - 2ad\)
Plugging in the given values:
- \(v_i^2 = (2.60 \, \text{m/s})^2 - 2a(40.0 \, \text{m})\)
We don't have the exact value for the acceleration, but we can use the approximation "approximately constant" as \(a\). So, the equation becomes:
- \(v_i^2 = (2.60 \, \text{m/s})^2 - 2(a)(40.0 \, \text{m})\)
Simplifying further, we have:
- \(v_i^2 = 6.76 \, \text{m}^2/\text{s}^2 - 80a \, \text{m}^2/\text{s}^2\)
Since we are only interested in the magnitude of the initial speed, we take the square root:
- \(v_i = \sqrt{6.76 \, \text{m}^2/\text{s}^2 - 80a \, \text{m}^2/\text{s}^2}\)
So, the car's original speed before braking is approximately \(v_i = \sqrt{6.76 \, \text{m}^2/\text{s}^2 - 80a \, \text{m}^2/\text{s}^2}\).
About equation of motion
The equation of motion is a mathematical expression that relates the initial velocity, final velocity, acceleration, displacement, and time of an object in motion. It describes the relationship between these variables and allows us to analyze the motion of an object under certain conditions.
The general equation of motion is given by:
\(s = ut + \frac{1}{2}at^2\)
Where:
- \(s\) is the displacement (change in position)
- \(u\) is the initial velocity
- \(t\) is the time elapsed
- \(a\) is the acceleration
This equation is valid when the acceleration is constant. It can be used to calculate the displacement of an object over a certain time interval or to determine the initial or final velocity of an object given the other variables.
There are also other equations of motion that involve different combinations of variables, such as the equation \(v = u + at\) which relates the initial velocity, final velocity, acceleration, and time. These equations are derived from the basic principles of kinematics and are widely used in physics and engineering to analyze the motion of objects.
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