A 34.0-kg crate is being pushed across a floor. The net horizontal force on the crate is 80.0N. Starting from rest, the crate is pushed over a distance of 4.4m. What is the final speed of the crate?
Well. Assuming no friction, a = F/m and v = sqrt(2ax)
To find the final speed of the crate, we can use the kinematic equation:
\(v^2 = u^2 + 2as\)
Where:
\(v\) = final velocity (unknown)
\(u\) = initial velocity (0 m/s, since the crate starts from rest)
\(a\) = acceleration
\(s\) = distance
First, let's find the acceleration. We can use Newton's second law:
\(F = ma\)
Where:
\(F\) = net force (80.0 N)
\(m\) = mass of the crate (34.0 kg)
Rearranging the equation to solve for \(a\), we have:
\(a = \frac{F}{m}\)
Substituting the given values:
\(a = \frac{80.0\, \text{N}}{34.0\, \text{kg}}\)
Now, we can calculate the acceleration:
\(a \approx 2.35\, \text{m/s}^2\)
Next, we can substitute the values of \(u\), \(a\), and \(s\) into the kinematic equation:
\(v^2 = (0\, \text{m/s})^2 + 2(2.35\, \text{m/s}^2)(4.4\, \text{m})\)
Simplifying:
\(v^2 = 20.62\, \text{m}^2/\text{s}^2\)
To find \(v\), we can take the square root of both sides:
\(v = \sqrt{20.62}\, \text{m/s}\)
Calculating:
\(v \approx 4.54\, \text{m/s}\)
Therefore, the final speed of the crate is approximately \(4.54\, \text{m/s}\).
To find the final speed of the crate, we can use the concept of work and energy.
The work done on an object is equal to the force applied multiplied by the distance over which the force is applied. In this case, the force is the net horizontal force of 80.0N, and the distance is 4.4m.
The work done on the crate can be calculated using the formula:
Work = Force * Distance
Work = 80.0N * 4.4m
Work = 352.0J (Joules)
The work done on the crate is equal to the change in kinetic energy of the crate.
The change in kinetic energy can be calculated using the formula:
Change in kinetic energy = (1/2) * mass * (final velocity^2 - initial velocity^2)
Since the crate starts from rest, the initial velocity is 0.
Therefore, the formula simplifies to:
Change in kinetic energy = (1/2) * mass * (final velocity^2)
We can rearrange the formula to solve for the final velocity:
final velocity^2 = (2 * Change in kinetic energy) / mass
Plugging in the values, we have:
final velocity^2 = (2 * 352.0J) / 34.0kg
final velocity^2 = 20.7059 m^2/s^2
Taking the square root of both sides, we find:
final velocity = √(20.7059 m^2/s^2)
final velocity ≈ 4.55 m/s
Therefore, the final speed of the crate is approximately 4.55 m/s.