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Kinetic energy and work done are related to each other as work done by a body is equal to a change in kinetic energy of the body.
- Kinetic energy is a type of mechanical energy that a particle or system possesses due to its motion.
- When work (energy transfer) is done on a body by applying a net force, the body accelerates and acquires kinetic energy.
- Kinetic energy is a property of a moving particle or system that is determined not only by its motion but also by its mass.
- This motion may be translational, vibrational, rotation about an axis, or any combination of these.
- The SI unit of kinetic energy is Joule (J).
For a body of mass m moving with velocity v, the formula of kinetic energy is given by
\(KE = \bigg(\frac{1}{2}\bigg)mv^2\)
Where
- KE is the kinetic energy of the body
- m is the mass of the body
- v is the velocity of the body
Work is the measure of energy transferred when a body is displaced along a length by an outside force, at least a portion of which is exerted in the same direction as the generated displacement.
- If the applied force is constant, the work done can be calculated by multiplying the length of the displaced path by the component of the applied force acting along the path.
- The SI unit of work is Joule (J).
The work done by a force is given by the formula
\(W = FS \; cosθ\)
Where
- F is the force acting on the body
- S is the displacement of the body
- θ is the angle between the force and displacement vector.
Very Short Answers Questions [1 Mark Questions]
Ques. Which of the following is correct?
- The kinetic energy of a system cannot be changed without changing the momentum
- The kinetic energy of a system can be changed without changing the momentum
- A system cannot have energy without having momentum
- The momentum of a system cannot be changed without changing its kinetic energy
Ans. The correct answer is b. The kinetic energy of a system can be changed without changing the momentum
Explanation: The kinetic energy of a system can be changed without affecting its momentum. A bomb explosion is a good example of this phenomenon. When a bomb explodes, the kinetic energy increases dramatically, yet the momentum remains constant. This may be demonstrated using vector algebra by obtaining the vector sum of all the velocities in the explosion.
Ques. Which of the following types of energy relies on the square of the body’s speed and mass?
- Nuclear energy
- Potential energy
- Electrostatic energy
- Kinetic energy
Ans. The correct answer is d. Kinetic energy
Explanation: The kinetic energy, nuclear energy, and gravitational potential energy of objects are all substantially influenced by their mass. Kinetic energy, on the other hand, is the only quantity that has the term square of velocity.
Ques. Work done in a system is
- The measure of energy change
- The measure of force
- The measure of displacement
- The measure of useful work
Ans. The correct answer is a. The measure of energy change
Explanation: Work done in any system is a measure of the energy change in the system with its surroundings. In practice, work done and energy change are the same. When a force is applied to a body, it moves and does some work. The amount of work done represents the energy change of the system with its surroundings.
Ques. Work being a scalar quantity sometimes has negative and positive signs. What does the sign suggest?
- The direction of work
- The direction of displacement
- Relative directions of force and displacement
- The direction of force
Ans. The correct answer is c. Relative directions of force and displacement
Explanation: Because work is a scalar quantity, it cannot have a direction. However, work is calculated by taking the dot product of force and displacement. The dot product is positive if both are pointing in the same direction. The dot product is negative if both are pointing in opposing directions. As a result, the sign of work denotes the relative direction of force and displacement.
Ques. When a body is falling freely under the influence of gravity, the kinetic energy remains constant.
- True
- False
Ans. The correct answer is b. False
Explanation: When a body falls freely under the effect of gravity, the kinetic energy varies as velocity changes, and the potential energy changes as height changes, but the total mechanical energy remains constant due to the law of conservation of energy.
Ques. Work is a
- Dimensionless quantity
- Vector quantity
- Scalar quantity
- Magnitude less quantity
Ans. The correct answer is c. Scalar quantity
Explanation: The physical quantity work only has magnitude but no sense of direction, therefore it is referred to as scalar quantity.
Short Answers Questions [2 Marks Questions]
Ques. What is kinetic energy?
Ans. Kinetic energy is a form of energy that an object or particle has due to its motion. Kinetic energy is the property of a moving object or particle that is affected by both its motion and its mass.
Ques. What is meant by work done?
Ans. The product of the force and the distance over which the force is applied is defined as work done. Work is completed when a force is applied to an object and the object moves across a distance. For example, when you lift an object, you exert a force over a distance and hence perform work.
Ques. What are the characteristics of kinetic energy?
Ans. The following are the characteristics of the kinetic energy
- The frame of reference affects kinetic energy.
- The kinetic energy of a body is always positive.
- Even when the amount and direction of force vary, the formula for kinetic energy, KE = \(\frac{1}{2}\) mv2, remains valid. As a result, the equation is true regardless of how the body gets velocity.
Ques. What is the formula of kinetic energy?
Ans. For a body of mass m moving with velocity v, the formula of kinetic energy is given by
\(KE = \bigg(\frac{1}{2}\bigg)mv^2\)
Where
- KE is the kinetic energy of the body
- m is the mass of the body
- v is the velocity of the body
Ques. What is the formula for work done by a body?
Ans. The work done by a force is given by the formula
\(W = FS \;cosθ\)
Where
- F is the force acting on the body
- S is the displacement of the body
- θ is the angle between the force and displacement vector.
Read More:
| Relevant Articles | ||
|---|---|---|
| Potential Energy Formula | Derivations of Kinetic Energy | Kinetic Theory |
| Relation between Kinetic Energy and Momentum | Kinetic and Potential Energy Difference | Rotational Kinetic Energy |
Long Answers Questions [3 Marks Questions]
Ques. An object of mass 10 kg is moving with a momentum of 15 kg m/s. A force of 0.2 N is applied to it in the direction of the motion for 10 sec. What will be the increase in kinetic energy?
Ans. Given
- Mass of the object, m = 10 kg
- Initial momentum, p = 15 kg m/s
- Force applied, F = 0.2 N
- Time, t = 10 seconds
Initial velocity of the object, u = p/m = \(\frac{15}{10}\) = 1.5 m/s
Initial kinetic energy, Ki = \(\frac{1}{2}\) mu2 = \(\frac{1}{2}\) x 10 x 1.52 = 11.25 J
Due to the applied force, the acceleration produced on the body,
a = F/m = \(\frac{0.2}{10}\) = 0.02 m/s2
Using the equation of motion, the final velocity of the body is given by
v = u + at = 1.5 + (0.02 x 10) = 1.7 m/s
Final kinetic energy, Kf = \(\frac{1}{2}\) mv2 = \(\frac{1}{2}\) x 10 x 1.72 = 14.45 J
Change in kinetic energy, ΔK = Final kinetic energy - Initial kinetic energy
⇒ ΔK = 14.45 - 11.25 = 3.2 J
Ques. Give the relationship between work done and kinetic energy.
Ans. The relationship between work and kinetic energy is given by the work-energy theorem.
It states that the work done by all forces acting on a particle is equal to the change in the kinetic energy of the particle.
Mathematically it can be represented as
WTOTAL = ΔKE = \(\bigg(\frac{1}{2}\bigg)\)mvf2 - \(\bigg(\frac{1}{2}\bigg)\)mvi2
Where
- W is the total work done measured in joules.
- m is the mass of the body calculated in kilograms.
- vi is the initial velocity of the body measured in m/s.
- vf is the final velocity of the object measured using m/s.
Ques. A body moves a distance of 10 m along a straight line under the action of a force of 5 N. If the work done is 25 J, then what is the angle that the force makes with the direction of motion of the body?
Ans. Given
- Distance traveled by the body, S = 10 m
- Force acting on the body, F = 5 N
- Work done, W = 25 J
Work done by a body is given by
W = FS cosθ
⇒ cosθ = W/FS
⇒ cosθ = \(\frac{25}{(5 \times 10)} = \frac{1}{2}\)
⇒ θ = 60°
Very Long Answers Questions [5 Marks Questions]
Ques. A shooter fires a bullet of mass 200 g with a speed of 200 m/s on soft plywood of thickness 5 cm. The bullet emerges with 15% of its initial kinetic energy. Find the emergent speed of the bullet.
Ans. Given
- Mass of the bullet, m = 200 g = 200 x 10-3 kg = 0.2 kg
- The initial speed of the bullet, vi = 200 m/s
The initial kinetic energy of the bullet, Ki = \(\frac{1}{2}\) mvi2
⇒ Ki = \(\frac{1}{2}\) x 0.2 x 2002 = 4000 J
Now, it is given that the bullet emerges with 15% of its initial kinetic energy. Therefore, the final kinetic energy of the bullet is 15% of the initial kinetic energy.
⇒ Final kinetic energy, Kf = 15% of Ki
⇒ Kf = \(\bigg(\frac{15}{100}\bigg)\) x 4000 = 600 J
Let vf be the final velocity (emergent speed) of the bullet, then the equation for the final kinetic energy is given by
Kf = \(\frac{1}{2}\) mvf2
⇒ vf = √(2Kf/m) = √(2 x 600 / 0.2) = 77.46 m/s
Hence, the emergent speed of the bullet is 77.46 m/s.
Ques. Derive the formula for the kinetic energy of an object of mass m, moving with velocity v.
Ans. Consider a body of mass m is moving in a straight line with acceleration a. After traveling distance S, starting from the rest let it acquire velocity v.
From the equation of motion, we have
v2 = u2 + 2aS
⇒ a = (v2 - u2)/2S = (v2 - 0)/2S
⇒ a = v2/2S ….(i)
Force acting on the body in the direction of displacement is given by
F = ma
Using equation (i), we get
F = m(v2/2S) ….(ii)
Work done by the force is given by
W = FS cosθ
Where θ is the angle formed by the force and displacement vectors. Because force and displacement are both in the same direction, θ = 0.
⇒ W = FS cos0 = FS
Using equation (ii), we get
W = m(v2/2S) x S = mv2/2 = 1/2 mv2
This work appears as kinetic energy in the body, so
Kinetic energy, KE = 1/2 mv2
Ques. The distance x moved by a body of mass 0.5 kg under the action of a force varies with time t as x(m) = 3t2 + 4t + 5.
Here, t is expressed in second. What is the work done by the force in the first 2 seconds?
Ans. The given equation is
x(m) = 3t2 + 4t + 5
Differentiating both sides with respect to time, we get
dx/dt = d/dt (3t2 + 4t + 5)
⇒ dx/dt = 6t + 4
But dx/dt = v, velocity
⇒ v = 6t + 4
Again differentiating with respect to time, we get
dv/dt = 6
Also, dv/dt = a, acceleration
⇒ a = 6 m/s2
Force, F = ma
⇒ F = 0.5 x 6 = 3 N
Now for the first 2 seconds, a positive sign in force and velocity shows that the angle between force and displacement is 0. Therefore
Work done, W = FS cos0
The displacement in the first 2 seconds is given by
S = x2 - x0 = [3(2)2 + 4(2) + 5] - [3(0)2 + 4(0) + 5]
⇒ S = 20 m
Therefore, work done, W = FS cosθ = 3 x 20 x 1 = 60 J
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