To determine the kinetic energy of the block after a horizontal force is applied, we first need to analyze the force function and how it affects the block's motion. The force acting on the block is given by the equation \( F = 2.5 - x^2 \), where \( x \) is the position of the block in meters. Since the block starts from rest, we can use the work-energy principle to find its kinetic energy.
Understanding the Force Function
The force function \( F = 2.5 - x^2 \) indicates that the force changes with the position of the block. Initially, when \( x = 0 \), the force is:
F(0) = 2.5 - (0)^2 = 2.5 N
As the block moves along the x-axis, the force will decrease as \( x \) increases, since the \( -x^2 \) term becomes more significant. The force will become zero when \( x = \sqrt{2.5} \approx 1.58 \) m, and it will become negative beyond this point, indicating that the block will experience a deceleration.
Calculating Work Done on the Block
The work done on the block as it moves from its initial position \( x = 0 \) to a position \( x = d \) can be calculated using the integral of the force over that distance:
W = ∫ F dx
For our case, we need to integrate from \( x = 0 \) to \( x = d \):
W = ∫ (2.5 - x^2) dx from 0 to d
Calculating this integral:
- First, find the antiderivative: \( ∫ (2.5 - x^2) dx = 2.5x - \frac{x^3}{3} + C \)
- Now, evaluate it from 0 to \( d \):
W = [2.5d - \frac{d^3}{3}] - [2.5(0) - \frac{(0)^3}{3}] = 2.5d - \frac{d^3}{3}
Relating Work to Kinetic Energy
According to the work-energy theorem, the work done on the block is equal to the change in kinetic energy:
W = ΔKE = KE_f - KE_i
Since the block starts from rest, \( KE_i = 0 \), so:
KE_f = W = 2.5d - \frac{d^3}{3}
Finding the Kinetic Energy at a Specific Position
To find the kinetic energy at a specific position, you can substitute the value of \( d \) into the equation. For example, if we want to find the kinetic energy when the block has moved 1 meter:
KE_f = 2.5(1) - \frac{(1)^3}{3} = 2.5 - \frac{1}{3} = 2.5 - 0.333 = 2.167 J
Final Thoughts
Thus, the kinetic energy of the block after it has moved 1 meter from its initial position is approximately 2.167 joules. This analysis illustrates how the force applied to the block not only influences its motion but also directly relates to its kinetic energy through the work done on it. Understanding these concepts is crucial in physics, as they form the foundation for analyzing motion and energy in various systems.
