19. In Figure 10-1, if the force exerted on a 3.0-kg backpack that is initally at rest is 20.0 N and the distance it acts over is 0.25 m, what is the final speed of the backpack?
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To find the final speed of the backpack, we can use the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
First, we need to calculate the work done on the backpack. The work done (W) is equal to the force (F) multiplied by the distance (d) over which it acts:
W = F * d
Plugging in the values given in the question, we have:
W = 20.0 N * 0.25 m
W = 5.0 J (Joules)
Next, we need to calculate the change in kinetic energy. The change in kinetic energy (ΔKE) is equal to the final kinetic energy (KE) minus the initial kinetic energy (KE₀).
Since the backpack is initially at rest, its initial kinetic energy is 0 (KE₀ = 0). The final kinetic energy (KE) can be calculated using the formula:
KE = 0.5 * mass * (velocity)²
In this case, the mass of the backpack is given as 3.0 kg. We need to solve for the velocity (final speed).
ΔKE = KE - KE₀
ΔKE = KE - 0
ΔKE = KE
Now, equating the work done to the change in kinetic energy, we have:
W = ΔKE
Plugging in the values:
5.0 J = (0.5 * 3.0 kg * (velocity)²) - 0
Simplifying the equation:
5.0 J = 1.5 kg * (velocity)²
Dividing both sides by 1.5 kg:
(velocity)² = 5.0 J / 1.5 kg
(velocity)² = 3.333...
Taking the square root of both sides:
velocity = √(3.333...) m/s
velocity ≈ 1.83 m/s
So, the final speed of the backpack is approximately 1.83 m/s.