All questions of Motion in Two Dimensions for Grade 9 Exam

As there is no horizontal acceleration is present in oblique projection horizontal component of velocity is constant

Centrifugal force, a fictitious force, peculiar to a particle moving on a circular path, that has the same magnitude and dimensions as the force that keeps the particle on its circular path (the centripetal force) but points in the opposite direction.

Given, mass of stone, m = 5 kg
Length of string, l = 10 m
Maximum tension, T = 160 N
We need to find the maximum velocity of revolution that can be given to the stone without breaking the string.

Maximum velocity of revolution:
We know that the tension in the string, T is given by:
T = (mv²) / r
where m is the mass of the stone, v is the velocity of the stone and r is the radius of the circle.

Finding the radius:
The length of the string, l is equal to the circumference of the circle, so:
l = 2πr
=> r = l / (2π)
=> r = 10 / (2π) = 1.59 m

Finding the maximum velocity:
Substituting the values in the tension equation, we get:
T = (mv²) / r
=> v² = (Tr) / m
=> v = √[(Tr) / m]

Now, substituting the given values, we get:
v = √[(160 × 1.59) / 5]
=> v = 17.88 m/s

Therefore, the maximum velocity of revolution that can be given to the stone without breaking the string is 17.88 m/s.

A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. ... The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

Understanding Projectiles
In physics, a projectile is an object that is thrown into the air with an initial velocity and is primarily influenced by the force of gravity and air resistance. To better understand why option 'C' (Taking off of an aircraft) is not a projectile, let's break down the examples.
Examples of Projectiles
- A bullet fired from a gun: Once fired, the bullet travels through the air under the influence of gravity and air resistance, making it a classic example of a projectile.
- A kicked football: When a football is kicked, it follows a curved trajectory due to the forces acting on it, including gravity, making it a projectile as well.
- A javelin thrown by an athlete: The javelin, once released, travels through the air in a parabolic path, demonstrating the characteristics of a projectile.
Why is Taking Off of an Aircraft Not a Projectile?
- Initial Lift-Off: When an aircraft takes off, it generates lift through its wings, which counteracts gravity. This lift is produced by the thrust from the engines, unlike projectiles that rely solely on their initial velocity and gravity.
- Controlled Flight: An aircraft is actively controlled during takeoff and flight, which means it is not simply following a path determined by gravity and air resistance. Instead, it relies on aerodynamic principles and engine thrust.
- Continuous Force Application: Projectiles do not have any forces acting on them after they are launched, except for gravity and air resistance. In contrast, aircraft are powered and maintained in the air by continuous engine thrust and aerodynamic lift.
In conclusion, while bullets, kicked footballs, and javelins follow the characteristics of projectiles, aircraft rely on lift and thrust, distinguishing them from true projectiles.

Circular motion involves the motion of an object in a circular path. The object moves around the center of the circle with a constant speed. In circular motion, the object experiences various types of acceleration and velocity. Let’s discuss the given options to understand circular motion in detail.

Radial acceleration is non-zero:
Radial acceleration is the acceleration of an object towards the center of the circle. In circular motion, the direction of velocity changes continuously, and hence the object experiences a change in direction. This change in direction leads to the acceleration of the object towards the center of the circle, known as radial acceleration. Therefore, option ‘a’ is correct.

Radial velocity is zero:
Radial velocity is the velocity of an object in the radial direction, i.e., towards the center of the circle. In circular motion, the object moves in a circular path without any radial motion, i.e., the object moves perpendicular to the radius of the circle. Hence, radial velocity is zero in circular motion. Therefore, option ‘b’ is incorrect.

Body is in equilibrium:
Equilibrium is a state where the object is at rest or moves with a constant velocity. In circular motion, the object moves with a constant speed, but the direction of the velocity changes continuously. Therefore, the object is not at rest, but it is also not moving with a constant velocity. Hence, the body is not in equilibrium in circular motion. Therefore, option ‘c’ is incorrect.

All of the above:
From the above discussion, we can conclude that radial acceleration is non-zero, radial velocity is zero, and the body is not in equilibrium in circular motion. Therefore, option ‘d’ is the correct answer.

Conclusion:
Circular motion involves the motion of an object in a circular path. The object experiences various types of acceleration and velocity in circular motion. In circular motion, radial acceleration is non-zero, radial velocity is zero, and the body is not in equilibrium.

Understanding the Problem
To find the time taken by the bird to cross the train, we first need to determine the relative speeds of the train and the bird.
Step 1: Convert Speeds to m/s
- The train's speed: 72 km/h = (72 * 1000) / (60 * 60) = 20 m/s
- The bird's speed: 18 km/h = (18 * 1000) / (60 * 60) = 5 m/s
Step 2: Calculate Relative Speed
Since the bird is flying in the opposite direction to the train, we add their speeds to find the relative speed:
- Relative speed = Speed of train + Speed of bird
- Relative speed = 20 m/s + 5 m/s = 25 m/s
Step 3: Calculate Time to Cross the Train
The time taken to cross the train is determined by the length of the train and the relative speed:
- Length of the train = 175 m
- Time = Distance / Speed
- Time = 175 m / 25 m/s = 7 seconds
Conclusion
Therefore, the time taken by the bird to cross the train is 7 seconds, which corresponds to option 'D'.
This exercise demonstrates the importance of understanding relative motion in physics, especially in problems involving objects moving in opposite directions.

Understanding Banked Curves
When trains navigate curves, banking the tracks is essential for safety and efficiency. This design helps manage the forces acting on the train as it travels around a curve.
Why Are Tracks Banked?
- Centripetal Force: When a train goes around a curve, it requires a centripetal force to change direction. This force is perpendicular to the train's motion and directed towards the center of the curve.
- Horizontal Component of Reaction: The banking of the tracks creates an angle between the vertical and the track surface. This angle allows the normal reaction force from the track to have a horizontal component that provides the necessary centripetal force.
Benefits of Banking
- Reduced Lateral Friction: By banking the tracks, the need for friction between the train's wheels and the track is minimized. This reduces wear and tear on both the train and the tracks.
- Enhanced Stability: Properly banked curves allow trains to maintain higher speeds without the risk of derailing, as the forces are balanced effectively.
- Improved Comfort: Passengers experience less lateral acceleration, making the ride more comfortable.
Conclusion
In summary, the statement that railway tracks are banked at curves to obtain the necessary centripetal force from the horizontal component of the reaction is indeed true (option 'A'). This engineering design is crucial for the safe and efficient operation of trains on curved tracks.

Explanation:
Non-uniform circular motion is the motion of an object moving in a circular path with a varying speed. The direction of the velocity and acceleration of an object in non-uniform circular motion changes at every point on the circular path.

Velocity:
Velocity is the rate of change of displacement with respect to time. In non-uniform circular motion, the velocity of the object is always tangent to the circular path. The magnitude of the velocity changes at every point on the circular path.

Acceleration:
Acceleration is the rate of change of velocity with respect to time. In non-uniform circular motion, the acceleration of the object is not constant. It changes at every point on the circular path.

Now, let's discuss the given options one by one:

a) Velocity is radial and acceleration is transverse only:
In this option, the velocity is radial which means it is directed towards the center of the circular path. But the acceleration is transverse which means it is perpendicular to the velocity vector. This option is correct because the direction of the acceleration is always towards the center of the circular path in non-uniform circular motion.

b) Velocity is transverse and acceleration is radial only:
In this option, the velocity is transverse which means it is perpendicular to the radius vector. But the acceleration is radial which means it is directed towards the center of the circular path. This option is incorrect because the direction of the acceleration is not always radial in non-uniform circular motion.

c) Velocity is radial and acceleration has both radial and transverse components:
In this option, the velocity is radial which means it is directed towards the center of the circular path. But the acceleration has both radial and transverse components. This option is incorrect because the direction of the acceleration is always towards the center of the circular path in non-uniform circular motion.

d) Velocity is transverse and acceleration has both radial and transverse components:
In this option, the velocity is transverse which means it is perpendicular to the radius vector. But the acceleration has both radial and transverse components. This option is incorrect because the direction of the acceleration is not always radial in non-uniform circular motion.

Hence, the correct option is 'a' - Velocity is radial and acceleration is transverse only.

Understanding Uniform Circular Motion
Uniform circular motion refers to motion along a circular path at a constant speed. In this scenario, the object’s speed remains constant, but its direction changes continuously, resulting in acceleration towards the center of the circle.
Analyzing the Options
- Motion of a racing car on a circular track
- This motion is often not uniform because a racing car changes its speed while navigating turns. Drivers accelerate or decelerate, which means the speed is not constant.
- Motion of the moon around the earth
- The moon moves in a nearly circular orbit at a constant speed, making this a classic case of uniform circular motion.
- Motion of a toy train on a circular track
- If the toy train moves at a constant speed around the track, it exemplifies uniform circular motion.
- Motion of the seconds hand on the circular dial of a watch
- The seconds hand moves at a constant speed, completing a full revolution in a fixed time, thus representing uniform circular motion.
Conclusion
The correct answer is option 'A' – the motion of a racing car on a circular track. This is because the car typically experiences variations in speed, unlike the other options, which maintain a constant speed while moving in a circular path.

**Uniform Circular Motion**

Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed. In this type of motion, the object moves along the circumference of the circle, maintaining a fixed distance from the center.

**Acceleration in Uniform Circular Motion**

While the speed of the object remains constant in uniform circular motion, it is important to note that the object is still experiencing acceleration. This may seem counterintuitive since we typically associate acceleration with a change in speed. However, in uniform circular motion, the acceleration is directed towards the center of the circle and is known as centripetal acceleration.

**Direction of Motion Changes**

The correct answer to why uniform circular motion is called continuously accelerated motion is that the direction of motion changes. In uniform circular motion, an object continuously changes its direction as it moves along the circular path. This change in direction implies that the object is experiencing acceleration.

**Velocity Remains the Same**

While the object is undergoing acceleration in uniform circular motion, its velocity remains constant. Velocity is a vector quantity that includes both magnitude (speed) and direction. In uniform circular motion, the speed of the object remains constant, but its direction changes. Therefore, the velocity of the object remains the same in terms of magnitude, but the direction of the velocity vector continually changes.

**Centripetal Acceleration**

The centripetal acceleration is the acceleration experienced by an object undergoing uniform circular motion. It is directed towards the center of the circle and is responsible for continuously changing the direction of the object's motion. The magnitude of the centripetal acceleration can be calculated using the formula:

a = v^2 / r

where a is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.

**Conclusion**

In conclusion, uniform circular motion is called continuously accelerated motion because the direction of motion of the object undergoing circular motion continually changes. Although the speed (magnitude of velocity) remains constant, the object experiences centripetal acceleration directed towards the center of the circle. This acceleration is responsible for continuously changing the direction of the object's motion, making it an example of accelerated motion.

Perpendicular Force on a Particle in a Plane

When a particle is acted upon by a force of constant magnitude that is always perpendicular to its velocity, the following can be observed:

Circular Motion

The particle moves in a circular path because the force acts as a centripetal force, pulling the particle towards the center of the circle.

Constant Kinetic Energy

Since the force is always perpendicular to the velocity of the particle, it does not do any work on the particle. Therefore, the kinetic energy of the particle remains constant.

Variable Speed

Although the particle moves in a circular path, its speed is not constant. This is because the force only changes the direction of the particle, not its speed. Therefore, the particle will move faster when it is farther away from the center of the circle and slower when it is closer to the center.

Conclusion

In conclusion, when a particle is acted upon by a force of constant magnitude that is always perpendicular to its velocity, it will move in a circular path with variable speed while maintaining constant kinetic energy.