Question

A particle of mass \(0.2 \mathrm{~kg}\) is moving in one dimension under a force that delivers a constant power \(0.5 W\) to the particle. If the initial speed (in \(m s^{-1}\) ) of the particle is zero, the speed (in \(m s^{-1}\) ) after \(5 s\) is _____ .

Short answer

\(v = \sqrt{25} = 5 \, m s^{-1}\)

Step-by-step solution

Understanding the Problem

We are given the constant power delivered to a particle, its mass, and the time duration during which the power is delivered. We need to find the final speed of the particle after 5 seconds assuming it started from rest.

Relating Power to Work and Energy

Power is the rate at which work is done. The work done on the particle equals the change in kinetic energy. Since the initial speed is zero, the work done will equal the final kinetic energy of the particle.

Using the Work-Energy Principle

Use the work-energy principle, which states that the work done on an object is equal to the change in kinetic energy. Since the initial kinetic energy is zero, the work done (Power × Time) will be equal to the final kinetic energy (\(\frac{1}{2}mv^2\)).

Calculating Final Kinetic Energy

Calculate the final kinetic energy using the power and time: Work Done = Power × Time = 0.5 W × 5 s = 2.5 J.

Computing the Final Speed

Final Kinetic Energy = \(\frac{1}{2}mv^2\). Solve for velocity v: \(v = \sqrt{\frac{2\times \text{Final Kinetic Energy}}{m}} = \sqrt{\frac{2\times 2.5}{0.2}}\).

Final Calculation

Perform the final calculation to find the speed after 5 seconds: \(v = \sqrt{\frac{5}{0.2}}\).

Key concepts

Understanding the Work-Energy Principle

The work-energy principle is a crucial concept in physics that connects force, motion, and energy. It essentially tells us that the work done by forces acting on an object results in a change in the object's kinetic energy.

Putting this into practice, if we apply a constant force to move an object in one dimension, the work done by that force is equal to the change in the object's kinetic energy. For a particle starting from rest, like in our exercise, this means that all the work done on it by external forces (in this case, due to the power being delivered) transforms into kinetic energy, causing the particle to accelerate.

• Work Done (W) = Change in Kinetic Energy (∆KE)
• W = Force (F) × Displacement (s) = Power (P) × Time (t)

So, with the power (P) and the time (t) provided, we can calculate the amount of work done, which is then used to find the particle's final kinetic energy and, consequently, its speed.

Calculating Kinetic Energy

Kinetic energy represents the energy an object possesses due to its motion. It can be calculated using the mass of the object and its speed. The formula to determine the kinetic energy (KE) of an object is:
\[KE = \frac{1}{2}mv^2\]
where:

• \(m\) is the mass of the object,
• \(v\) is the velocity of the object.

In our example, after using the power provided and the time duration, we find the work done which is equal to the final kinetic energy (since the initial kinetic energy is zero). That allows us to solve for the particle's velocity after the given time.

Relationship Between Power and Speed

Power in physics is defined as the rate at which work is done or energy is transferred. It can be expressed as:
\[P = \frac{W}{t} = Fv\]
where:

• \(P\) stands for power,
• \(W\) is work done,
• \(t\) is time,
• \(F\) is force, and
• \(v\) is the velocity or speed of the object.

Thus, if the power transferred to an object is known, and the force acting upon it remains constant, we can infer the object's speed. This relationship is fundamental when analyzing situations where power output and speed changes are involved, like an engine of a car accelerating or, as in our exercise, a particle being propelled by a constant power source.

Force and Motion in One Dimension

When dealing with force and motion in one dimension, we use Newton's second law of motion which states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration (F = ma).

In scenarios where objects move along a straight line, such as the particle in our textbook problem, analyzing the motion becomes simpler. We equate the rate of change of kinetic energy to the work done by the force (because work is defined as the force times displacement). Here, power delivering a constant force results in a constant acceleration and uniform change in speed over time. This direct relationship makes it easier to translate the power delivered into a final speed, as demonstrated in the problem where the speed after 5 seconds is solely based on the constant power applied to the particle.