91Ó°ÊÓ

Kinetic energy is a form of energy that an object possesses due to its motion. Imagine it as the energy that is required to accelerate an object from rest to its current velocity. The more massive an object is and the faster it moves, the more kinetic energy it has. It is calculated using the formula:

• \( K = \frac{1}{2} mv^2 \)

where \( K \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) is the velocity of the object.
In this exercise, the canister experienced a change in kinetic energy as it changed its velocity due to the applied force. Initially, when it was moving at \(4.0 \, \mathrm{m/s}\), the canister had an initial kinetic energy \(K_i = 16.0 \, \mathrm{J}\). After speeding up to \(6.0 \, \mathrm{m/s}\) in another direction, its kinetic energy increased to \(K_f = 36.0 \, \mathrm{J}\).
This change signifies the amount of work done on the canister by the force acting on it.

Velocity vectors help us understand the direction and magnitude of an object's velocity. In physics, velocity is not just about how fast something moves, but also the direction in which it moves. Hence, we represent it as a vector, which contains information about both speed and direction.
In the problem, the canister's initial and final velocities can be described using velocity vectors:

• Initial Velocity Vector \( \mathbf{v_i} = (4.0 \, \mathrm{m/s}, 0) \)
• Final Velocity Vector \( \mathbf{v_f} = (0, 6.0 \, \mathrm{m/s}) \)

These vectors indicate a change in direction and speed, rooted in the concept of vectors in mathematics. Initially, the object moves straight along the x-axis, while later, it shifts to move along the y-axis. This alteration in motion is vectorially significant and impacts how we calculate changes in kinetic energy.

In simple terms, force in physics is an interaction that changes the motion of an object. It is what causes an object to start moving, stop moving, or alter its motion. Forces have both magnitude and direction, making them vector quantities closely linked to velocity vectors.
In our scenario, a force of magnitude \(5.0 \, \mathrm{N}\) is the external influence altering the canister’s velocity. This force causes the canister to change direction and speed over time, which in turn impacts its kinetic energy. Applying the work-energy theorem, the work done by this force is equal to the change in the object’s kinetic energy:

• \( W = K_f - K_i \)

Where \( W \) is the work done by the force, deriving from the measurable shift in kinetic energy from \(16.0 \, \mathrm{J}\) to \(36.0 \, \mathrm{J}\), resulting in \( W = 20.0 \, \mathrm{J}\). Understanding this, we see the profound connection between forces, work done, and kinetic energy changes.