When an object starts from rest and accelerates, it undergoes changes in its motion due to the acceleration. Acceleration refers to the rate of change of velocity, which means how quickly or slowly an object’s velocity is changing over time. An object at rest has zero velocity, so when it starts accelerating from rest, its velocity starts increasing from zero. The acceleration causes the velocity to continuously change as time passes.

### What is acceleration?

Acceleration is defined as the rate of change of velocity with respect to time. It refers to how quickly an object’s velocity is changing. The acceleration of an object can be positive or negative depending on whether the velocity is increasing or decreasing. Acceleration is a vector quantity, having both magnitude and direction. The SI unit of acceleration is meters per second squared (m/s2).

### What causes acceleration?

There are two main causes of acceleration:

- Applying an external force – When a net external force acts on an object, it causes the object to accelerate in the direction of the force. The acceleration is directly proportional to the net force and inversely proportional to the mass.
- Changing direction of motion – Even if the speed remains constant, a change in the direction of motion constitutes acceleration. This acceleration is called centripetal acceleration and its direction is towards the center of curvature of the path.

Common examples include the acceleration of a car due to the engine force or the acceleration of a stone due to gravity.

## Behavior of objects starting from rest

When objects start moving from rest, i.e. their initial velocity is zero, their motion under different conditions is as follows:

### Under constant acceleration

If an object starts from rest and has a constant acceleration a, its velocity v changes as:

v = at

Where t is the time elapsed.

Its displacement s also changes as:

s = (1/2)at^{2}

These equations show that under constant acceleration, the velocity increases linearly with time while the displacement increases quadratically with time.

### Under variable acceleration

If the acceleration of an object starting from rest keeps changing with time, its velocity and displacement change accordingly. The velocity at any instant is the integral of acceleration from start till that instant.

v = ∫a dt

The displacement is the integral of velocity from start till that instant.

s = ∫v dt

Thus, variable acceleration leads to nonlinear increase in velocity and displacement.

### Under gravity

When objects fall freely under gravity starting from rest, they undergo uniformly accelerating motion with acceleration g directed downwards. Here,

v = gt

s = (1/2)gt^{2}

The velocity increases linearly while displacement increases quadratically with time.

## Examples of objects starting from rest

### Free fall under gravity

When an object is dropped from some height or thrown vertically upwards, it starts from rest and undergoes free fall under gravity. It accelerates downwards at g = 9.8 m/s^{2}. Its velocity increases as gt while its displacement follows (1/2)gt^{2}.

### Motion on an inclined plane

When a ball rolls down an inclined plane starting from rest, its motion has an accelerating effect due to gravity. The acceleration is gsinθ where θ is the angle of inclination. As it accelerates down the slope, its velocity increases linearly while the displacement increases quadratically.

### Acceleration of vehicles

When a vehicle starts from rest and the driver steps on the accelerator, it accelerates forward due to the engine driving force. The rate of acceleration depends on the power of the engine. The velocity increases gradually from 0 and the vehicle covers larger and larger distances.

### Atwood machine

In an Atwood machine, two masses connected by a string move with accelerating motion when released from rest. Depending on the mass ratio, the acceleration of the masses can be calculated. Accordingly, the velocities and displacements increase with time.

## Measuring acceleration from rest

The acceleration of an object starting from rest can be measured in the following ways:

### Using sensors

Accelerometers and motion sensors can be attached to the object to directly measure the rate of change of velocity with time. Graphs of velocity versus time can be plotted.

### Frame-by-frame video analysis

Using high speed cameras and video analysis software, the positions of the object in successive frames starting from rest can be used to calculate velocity and acceleration.

### Motion equations

By measuring the time taken and distance travelled till the object attains a certain velocity starting from rest, acceleration can be calculated using equations of motion.

### Inclined planes

For objects accelerating down inclined planes, the acceleration can be calculated as gsinθ by measuring the angle of inclination θ.

### Collisions

Conservation of momentum and energy equations before and after collisions can be used to calculate the accelerations involved when objects start from rest.

## Significance of acceleration from rest

The study of acceleration of objects starting from rest has many useful applications and implications:

- It helps measure forces acting on an object based on its acceleration.
- It is useful in design of vehicle engines to achieve desired accelerations.
- Such motion forms the basis of understanding free fall, projectile motion and motion on inclined planes.
- It demonstrates the effect of unbalanced forces on objects starting from rest.
- It provides insights into building efficient momentum exchange devices like roller coasters and Atwood machines.
- It helps deduce information about resistive forces acting on objects.
- The study establishes the significance of inertia, which must be overcome to accelerate objects from rest.

Thus, analyzing the motion resulting from acceleration starting from rest has wide practical and theoretical relevance.

## Conclusion

When an object starts accelerating from rest, its velocity increases progressively as its displacement also changes accordingly. The exact change depends on whether the acceleration is constant, variable or under gravity. Acceleration from rest demonstrates Newton’s laws of motion and provides insights into various applications like vehicle engines, free fall motion, collisions, and inclined plane movement. Techniques like motion sensors, video analysis and equations of motion help measure acceleration starting from zero initial velocity. The study helps reveal the effects of unbalanced forces and inertia on bodies starting from rest.