QPage 48-49: Q5: HC Verma: Concepts of Physics - Volume 1 Chapter-3 Rest and Motion Kinematics Page 48-49: Q5 Solution: The velocity of a particle is towards west at an instant. Its…

Step 1: Restating the Conditions

At a given instant:

So the acceleration is not along any of the four cardinal directions (N, S, E, W).

Step 2: What are the possible directions left?

If acceleration is a vector in the horizontal plane, and it is not along N, S, E, or W, then it must be at some angle to these axes—for example:

1. North-East (NE)
2. North-West (NW)
3. South-East (SE)
4. South-West (SW)

Or more generally, any direction that is not exactly 0°, 90°, 180°, or 270° from the East direction.

So, the acceleration could be, say, towards North-West, South-East, etc.

Step 3: Connecting Velocity and Acceleration

Recall the key physics principle:

Acceleration is the rate of change of velocity.

If acceleration has a component perpendicular to velocity, it changes the direction of velocity (without changing speed).

If acceleration has a component parallel (or anti-parallel) to velocity, it changes the speed.

In this problem, we are told the velocity is due West. The acceleration is not exactly West, East, North, or South. That means the acceleration has both:

1. A component perpendicular to West (i.e., North or South) → changes direction.
2. A component parallel or anti-parallel to West (i.e., West or East) → changes speed.

So the motion is a combination of turning and speeding up/slowing down.

Step 4: A Perfect Example – Motion Along a Curved Path (Not a Perfect Circle)

The simplest and clearest example is:

A particle moving along a curved (non-circular) path, such as a parabolic trajectory or an ellipse, where at a particular instant its velocity is due West, and its acceleration points towards, say, North-West (NW) or South-West (SW).

Concrete Example 1: Projectile Motion at a Specific Instant

Consider a ball thrown upward and toward the East, but at the instant when it is at the topmost point of its trajectory and moving due West (say, because of wind or an initial westward component after reaching the peak).

At that instant:

1. Velocity = due West.
2. Acceleration = due downward (gravity).

But wait—downward is South (if we take North as up). That would be South, which is not allowed in the problem! So projectile motion with gravity alone won't work because gravity is vertical (North/South) and the problem says acceleration is not towards North or South.

So we need a motion where acceleration is in a diagonal direction.

Concrete Example 2: Motion on a Curved Track with Variable Speed

Imagine a car or a particle moving along a curved path (like a bend in a road) that is not a perfect circle. At a particular instant:

1. The car is heading due West.
2. The car is both turning and accelerating/braking.
3. The net acceleration is directed, say, towards South-West (SW).

South-West is:

1. Not purely West
2. Not purely East
3. Not purely North
4. Not purely South

So it satisfies the condition.

At that instant:

1. The South component of acceleration turns the car (changes direction).
2. The West component of acceleration changes the speed (if West = speeding up; if East = slowing down, but here it's West, so it's speeding up while turning).

Concrete Example 3: Particle in Uniform Circular Motion (with a twist)

If a particle moves in uniform circular motion at constant speed:

1. Velocity is always tangential (say, West at a given instant).
2. Acceleration is always radial (towards the center).

If the center of the circle is not exactly North, South, East, or West from the particle, but say at a 45° angle (North-East or South-East), then the acceleration points diagonally.

Example: A particle moving in a circle, and at the instant when it is at the leftmost point of the circle:

1. Velocity = due West (tangent to the circle).
2. The center of the circle is to the North-East or South-East (depending on the circle's position).

If the center is, say, North-East of the particle, then acceleration = towards North-East, which is:

1. Not West ❌
2. Not East ❌
3. Not North ❌ (it's NE, not purely North)
4. Not South ❌

So this fits perfectly!

Step 5: The Best and Simplest Answer

A particle moving in a circular path, where at the instant it is at the leftmost point of the circle, its velocity is due West, and the centre of the circle lies to the North-East (or South-East) of the particle. The acceleration (centripetal) is then directed towards the centre, i.e., North-East (or South-East), which is not purely North, South, East, or West.

Alternatively:

A particle moving along a curved track (like a roller coaster or a bend) at an instant when it is heading West, and the net acceleration (due to both turning and changing speed) points diagonally, say South-West or North-West.

Step 6: Why This Works

Final Answer

Example: A particle moves in a circular path. At the instant when it is at the leftmost point of the circle, its velocity is towards the West. If the centre of the circle is located to the North-East of the particle, then the centripetal acceleration is directed towards the North-East, which is neither West, East, North, nor South. This satisfies all the given conditions.