Exercise Solutions

Q.1 Multiple Choice Questions (Choose the correct option):

1. An object travels 10 m in 2 s and then another 10 m in 3 s. The average speed of the object is:
   a) 4 m/s
   b) 5 m/s
   c) 2 m/s
   d) 3 m/s
   Solution: Total Distance = $10 \text{ m} + 10 \text{ m} = 20 \text{ m}$ Total Time = $2 \text{ s} + 3 \text{ s} = 5 \text{ s}$ Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{20 \text{ m}}{5 \text{ s}} = 4 \text{ m/s}$
   Correct Option: a) 4 m/s
2. If an object starts from rest and accelerates uniformly at 2 m/s² for 5 s, its final velocity will be:
   a) 5 m/s
   b) 10 m/s
   c) 2 m/s
   d) 20 m/s
   Solution: Given: Initial velocity ($u$) = $0 \text{ m/s}$ (starts from rest) Acceleration ($a$) = $2 \text{ m/s}^2$ Time ($t$) = $5 \text{ s}$ Using the first equation of motion: $v = u + at$ $v = 0 + (2 \text{ m/s}^2 \times 5 \text{ s})$ $v = 10 \text{ m/s}$
   Correct Option: b) 10 m/s
3. Newton's first law of motion is also known as the law of:
   a) Momentum
   b) Action-reaction
   c) Inertia
   d) Force
   Solution: Newton's first law defines inertia, the tendency of an object to resist changes in its state of motion.
   Correct Option: c) Inertia
4. The unit of momentum is:
   a) N
   b) J
   c) kg m/s
   d) kg m/s²
   Solution: Momentum ($p$) = mass ($m$) $\times$ velocity ($v$). The unit of mass is kg and velocity is m/s. So, the unit of momentum is kg m/s.
   Correct Option: c) kg m/s
5. When a force of 1 N acts on a mass of 1 kg, the acceleration produced is:
   a) 1 m/s²
   b) 9.8 m/s²
   c) 10 m/s²
   d) 0.1 m/s²
   Solution: Given: Force ($F$) = $1 \text{ N}$ Mass ($m$) = $1 \text{ kg}$ Using Newton's Second Law: $F = ma$ $1 \text{ N} = 1 \text{ kg} \times a$ $a = 1 \text{ m/s}^2$
   Correct Option: a) 1 m/s²

Q.2 Fill in the blanks:

1. The rate of change of displacement is called velocity.
2. The acceleration due to gravity on Earth is approximately 9.8 m/s².
3. Newton's second law of motion relates force to the rate of change of momentum (or mass and acceleration).
4. Inertia is the property of an object to resist a change in its state of rest or motion.
5. For every action, there is an equal and opposite reaction.

Q.3 Match the pairs:

(Note: As an AI, I will provide the correct pairs.)

• 1. Velocity - c. m/s
• 2. Acceleration - b. m/s²
• 3. Momentum - a. kg m/s
• 4. Force - d. N

Q.4 True or False. If false, write the correct statement:

1. Distance is a vector quantity.
   False. Distance is a scalar quantity.
2. Negative acceleration is also called deceleration.
   True.
3. Newton's third law of motion applies to objects in rest.
   True. Newton's third law applies to all interactions, whether the objects are at rest or in motion. For example, a book resting on a table exerts a downward force (action), and the table exerts an equal and opposite upward force (reaction) on the book.
4. The mass of an object changes with its velocity.
   False. The mass of an object is a fundamental property and generally does not change with its velocity in classical mechanics. (Relativistic effects at very high speeds are not considered at this level).
5. The total momentum of an isolated system remains conserved.
   True.

Q.5 Answer the following questions:

1. Define motion. Give examples.
   Answer: Motion is defined as the change in position of an object with respect to a reference point (or observer) over time. It is a relative concept.
   Examples: A car moving on a road, a ball falling from a height, a person walking, the Earth revolving around the Sun.
2. Distinguish between distance and displacement.
   Answer:
   • Distance: It is the actual path length covered by an object. It is a scalar quantity (only magnitude). It can never be zero or negative during motion.
   • Displacement: It is the shortest straight-line path between the initial and final positions of an object, along with its direction. It is a vector quantity (magnitude and direction). It can be zero, positive, or negative.
3. What is acceleration? Explain its types.
   Answer: Acceleration is the rate of change of velocity of an object. It is a vector quantity.
   Types:
   • Positive Acceleration: Velocity increases in the direction of motion.
   • Negative Acceleration (Deceleration/Retardation): Velocity decreases in the direction of motion.
   • Zero Acceleration: Velocity remains constant (uniform velocity).
4. State Newton's first law of motion. Give two examples.
   Answer: Newton's First Law of Motion states that an object at rest will remain at rest, and an object in motion will remain in motion with the same speed and in the same direction unless acted upon by an unbalanced external force.
   Examples:
   1. When a vehicle suddenly starts, passengers tend to fall backward due to inertia of rest.
   2. When a moving vehicle suddenly stops, passengers tend to fall forward due to inertia of motion.
5. State Newton's second law of motion. Write its mathematical form.
   Answer: Newton's Second Law of Motion states that the rate of change of momentum of an object is directly proportional to the applied unbalanced force and takes place in the direction of the force.
   Mathematical Form: $F = ma$ (where $F$ is force, $m$ is mass, and $a$ is acceleration).
6. State Newton's third law of motion. Give two examples.
   Answer: Newton's Third Law of Motion states that for every action, there is always an equal and opposite reaction. These forces act on two different objects.
   Examples:
   1. When a person swims, they push the water backward (action), and the water pushes the person forward (reaction).
   2. When a gun is fired, the bullet moves forward (action), and the gun recoils backward (reaction).
7. Explain the law of conservation of momentum.
   Answer: The Law of Conservation of Momentum states that in an isolated system (a system where no external unbalanced force acts), the total momentum of the system remains constant. This means that the sum of the momenta of all objects in the system before an interaction (like a collision or explosion) is equal to the sum of their momenta after the interaction. Mathematically, for two colliding objects: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$.
8. What is free fall? What is the value of 'g'?
   Answer: Free fall is the motion of an object under the sole influence of gravity, neglecting air resistance. The acceleration experienced by an object in free fall is called the acceleration due to gravity, denoted by 'g'.
   The approximate value of 'g' on Earth is $9.8 \text{ m/s}^2$.
9. Derive the first equation of motion ($v = u + at$) graphically.
   Answer: Consider an object moving with uniform acceleration 'a'. Let its initial velocity at time $t=0$ be 'u' and its final velocity at time 't' be 'v'.
   On a velocity-time graph:
   - Plot time (t) on the x-axis and velocity (v) on the y-axis.
   - Since the acceleration is uniform, the velocity-time graph will be a straight line.
   - Let point A represent the initial velocity 'u' at $t=0$.
   - Let point B represent the final velocity 'v' at time 't'.
   - Draw a perpendicular from B to the time axis, meeting at C.
   - Draw a perpendicular from A to BC, meeting at D.
   - From the graph:
   - Initial velocity = OA = u
   - Final velocity = BC = v
   - Time interval = OC = t
   - Change in velocity = BD = BC - DC = BC - OA = v - u
   - We know that acceleration (a) is the slope of the velocity-time graph.
   - Slope = $\frac{\text{Change in velocity}}{\text{Time taken}} = \frac{BD}{AD}$
   - Since AD = OC = t, we have: $a = \frac{v - u}{t}$
   - Rearranging this equation, we get: $at = v - u$
   - Therefore, $v = u + at$. This is the first equation of motion.
10. What is uniform circular motion? Give an example.
    Answer: Uniform circular motion is the motion of an object moving in a circular path with a constant speed. Although the speed is constant, the direction of the object's velocity continuously changes at every point along the circle. Because the velocity's direction changes, the object is constantly accelerating (this acceleration is called centripetal acceleration).
    Example: The motion of the tip of a second hand of a clock, or the motion of a satellite orbiting the Earth at a constant altitude.

Q.6 Give reasons:

1. An object thrown upwards comes back down.
   Reason: An object thrown upwards comes back down due to the gravitational force of the Earth. Gravity constantly pulls all objects towards the center of the Earth, causing a downward acceleration ('g'). This force eventually overcomes the initial upward velocity, brings the object to a momentary halt at its peak, and then pulls it back down.
2. It is difficult to walk on a slippery road.
   Reason: It is difficult to walk on a slippery road because there is very little friction between our feet and the road surface. Friction is the force that opposes motion and allows us to push backward on the ground (action) to move forward (reaction). On a slippery road, the insufficient friction means we cannot exert enough action force, and thus the reaction force from the ground is too small to propel us forward effectively, leading to slipping.
3. A passenger standing in a bus falls backward when the bus starts suddenly.
   Reason: When a bus starts suddenly, the passenger's feet, being in contact with the bus floor, move forward with the bus. However, due to inertia of rest, the upper part of the passenger's body tends to remain in its state of rest. This resistance to change in motion causes the passenger to fall backward relative to the bus.
4. A karate player breaks a slab of ice with a single blow.
   Reason: A karate player breaks a slab of ice with a single blow by applying a large force over a very short interval of time. According to Newton's Second Law of Motion ($F = \frac{\Delta p}{\Delta t}$), a large force can be generated if the change in momentum ($\Delta p$) occurs in a very short time ($\Delta t$). The player delivers a sudden, powerful blow, imparting a large impulse (change in momentum) to the ice, which generates a very large force sufficient to break it.
5. A rocket moves upwards after ignition.
   Reason: A rocket moves upwards after ignition based on Newton's Third Law of Motion and the Law of Conservation of Momentum. The rocket expels hot gases downwards at a very high velocity (action). In response, these expelled gases exert an equal and opposite reaction force upwards on the rocket, propelling it into the air. This is also explained by conservation of momentum: the downward momentum of the gases is balanced by the upward momentum of the rocket.

Q.7 Solve the following numerical problems:

1. A car travels 50 km in the first hour, 60 km in the second hour, and 40 km in the third hour. Calculate its average speed.
   Solution: Total distance covered = $50 \text{ km} + 60 \text{ km} + 40 \text{ km} = 150 \text{ km}$ Total time taken = $1 \text{ hour} + 1 \text{ hour} + 1 \text{ hour} = 3 \text{ hours}$ Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{150 \text{ km}}{3 \text{ hours}} = 50 \text{ km/h}$
   The average speed of the car is $50 \text{ km/h}$.
2. A train starting from rest attains a velocity of 72 km/h in 5 minutes. Calculate the acceleration.
   Solution: Given: Initial velocity ($u$) = $0 \text{ m/s}$ (starts from rest) Final velocity ($v$) = $72 \text{ km/h}$ Convert $v$ to m/s: $72 \text{ km/h} = 72 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 72 \times \frac{5}{18} \text{ m/s} = 4 \times 5 \text{ m/s} = 20 \text{ m/s}$ Time ($t$) = $5 \text{ minutes}$ Convert $t$ to seconds: $5 \text{ minutes} = 5 \times 60 \text{ s} = 300 \text{ s}$ Using the first equation of motion: $v = u + at$ $20 = 0 + a \times 300$ $20 = 300a$ $a = \frac{20}{300} = \frac{2}{30} = \frac{1}{15} \text{ m/s}^2$ $a \approx 0.067 \text{ m/s}^2$
   The acceleration of the train is approximately $0.067 \text{ m/s}^2$.
3. A stone is dropped from a height of 20 m. Calculate the time it takes to reach the ground and its final velocity. (Assume g = 10 m/s²)
   Solution: Given: Initial velocity ($u$) = $0 \text{ m/s}$ (dropped) Height ($h$) = $20 \text{ m}$ (displacement $s = 20 \text{ m}$) Acceleration due to gravity ($g$) = $10 \text{ m/s}^2$ (downwards, so positive) To find time ($t$): Using the second equation of motion for free fall: $h = ut + \frac{1}{2}gt^2$ $20 = (0 \times t) + \frac{1}{2} \times 10 \times t^2$ $20 = 5t^2$ $t^2 = \frac{20}{5} = 4$ $t = \sqrt{4} = 2 \text{ s}$ (time cannot be negative) To find final velocity ($v$): Using the first equation of motion for free fall: $v = u + gt$ $v = 0 + (10 \text{ m/s}^2 \times 2 \text{ s})$ $v = 20 \text{ m/s}$
   The stone takes $2 \text{ s}$ to reach the ground, and its final velocity is $20 \text{ m/s}$.
4. A force of 10 N acts on an object of mass 2 kg for 3 seconds. Calculate the change in momentum of the object.
   Solution: Given: Force ($F$) = $10 \text{ N}$ Mass ($m$) = $2 \text{ kg}$ Time ($t$) = $3 \text{ s}$ We know that Force is the rate of change of momentum: $F = \frac{\Delta p}{\Delta t}$ So, Change in momentum ($\Delta p$) = $F \times \Delta t$ $\Delta p = 10 \text{ N} \times 3 \text{ s}$ $\Delta p = 30 \text{ kg m/s}$
   The change in momentum of the object is $30 \text{ kg m/s}$.
5. An object of mass 5 kg is moving with a velocity of 10 m/s. It collides with another object of mass 3 kg moving with a velocity of 5 m/s in the same direction. If they stick together after collision, what is their combined velocity?
   Solution: Given: Mass of first object ($m_1$) = $5 \text{ kg}$ Initial velocity of first object ($u_1$) = $10 \text{ m/s}$ Mass of second object ($m_2$) = $3 \text{ kg}$ Initial velocity of second object ($u_2$) = $5 \text{ m/s}$ Since they stick together, their final velocity will be the same, let it be $V$. Using the Law of Conservation of Momentum: Total momentum before collision = Total momentum after collision $m_1 u_1 + m_2 u_2 = (m_1 + m_2) V$ $(5 \text{ kg} \times 10 \text{ m/s}) + (3 \text{ kg} \times 5 \text{ m/s}) = (5 \text{ kg} + 3 \text{ kg}) V$ $50 \text{ kg m/s} + 15 \text{ kg m/s} = 8 \text{ kg} \times V$ $65 \text{ kg m/s} = 8 \text{ kg} \times V$ $V = \frac{65}{8} \text{ m/s}$ $V = 8.125 \text{ m/s}$
   Their combined velocity after collision is $8.125 \text{ m/s}$ in the original direction.