Motion

Yes, a body can have a constant speed and still be accelerating.

Acceleration can be either due to change in speed or due to direction of motion or both.

Consider an example of uniform circular motion. In a uniform circular motion, the body moves with a constant speed but we still say that it is accelerating due to the change in direction.

This acceleration is called centripetal acceleration and is given as

a_c =

Where, v = constant speed

And r = radius of circle

Displacement – Time graph of a body moving with constant acceleration is as shown in the right side.

Explanation:

Slope of displacement – time graph gives the velocity.

Consider the following graph:

Here AB = Displacement of Body

And OB = Time Taken

Now, Velocity =

But is known as slope of line OA.

We know that in constant acceleration, velocity is changing with time at a constant rate.

Since, Acceleration =

So, slope of displacement – time graph is changing at a constant rate and hence we get a graph like shown at the beginning.

No, speed of a body can never be negative. Speed is a scalar quantity given as:

Speed =

Distance travelled by a body is always positive. It may be zero, but is never negative.

Time taken is also positive quantity.

So, speed being the ratio of two positive quantities can never be negative.

The slope of velocity – time graph represent acceleration. Since, Slope =

= Change in y – axis

= Change in x – axis

y = Velocity = v

x = time = t

So, = = Change in velocity

= = Change in time

So, Slope = =

We know that Acceleration =

So, Slope of Velocity – Time graph is acceleration.

Distance travelled by the particle cannot be zero when displacement is not zero.

Distance is the actual length of the path covered by a moving body irrespective of the direction in which body trace.

When a body moves from one position to another, then shortest (straight line) distance between initial and final positions of the body along with direction is displacement.

The above mentioned relation implies that the distance is always greater than or equal to displacement.

Hence, if 0, then Distance 0

The athlete was expected to take four rounds such that: line of finish = line of start

This means, we are saying that athlete comes back to its initial position. This is only possible if there is change in direction.

Without the change in direction a person can never reach back the initial point.

Now, we know that change in direction means acceleration, hence motion is non uniform.

Odometer is used to measure the distance travelled by the vehicle.

Given:

Initial Reading = 1048 km at t = 0 seconds

Final Reading = 1096 km at t = 40 minutes = hr

Average Speed = = =

= 72 kmhr^-1

Calculations using the readings of odometer gives the average speed.

Whereas, the speedometer gives the reading of instantaneous speed.

Thus, the two values, that is one calculated and one on the speedometer may not be the same.

Both can only be same if the vehicle has been moving with constant velocity throughout the measurement.

Acceleration is the rate of change of velocity. In order to have non-zero acceleration, rate of change of velocity should be non-zero.

Yes, it is possible to have non-zero acceleration at zero velocity. But it is only possible for a moment. A moment is a brief period that cannot be measured.

Let us understand it by following two examples:

a) When we throw a ball in the air, it attains a maximum height and then comes down. At the maximum height velocity of the ball is zero, but a negative acceleration due to gravitation acts on it.

b) When we apply brakes on a moving car, the car stops and the velocity of the car is zero at that moment but acceleration (retardation) is not zero.

Yes, a body can move horizontally with acceleration in vertical direction.

Consider the application of centripetal acceleration on a body for an example.

When a body moves on a circular path, then centripetal acceleration is always in radial direction and towards the center.

So, consider a body moving on a vertical circle passing its highest and lowest point.

a_c acts vertically whereas v in both cases acts horizontally.

Yes, a body can have a constant speed and still be accelerating.

Acceleration can be either due to change in speed or due to direction of motion or both.

Consider an example of uniform circular motion. In a uniform circular motion, the body moves with a constant speed but we still say that it is accelerating due to the change in direction.

This acceleration is called centripetal acceleration and is given as

a_c =

Where, v = constant speed

And r = radius of circle

Displacement – Time graph is a plot of points telling the position of particles at that instant of time.

Y – Axis : Displacement

X – Axis : Time

We plot the points for different positions of particle at different times.

Slope of displacement – time graph gives the velocity.

Consider an example of a girl going to school with uniform velocity.

Uniform velocity means equal displacement in equal intervals of time. Thus for uniform velocity the graph of displacement against time will be straight line as shown below:

To calculate uniform velocity from the graph,

The slope of displacement – time graph indicates velocity of the body. So, the slope of the displacement – time graph can be used to calculate the velocity of the body.

For calculating the slope of the displacement – time graph, we take a point A on the straight line graph and drop a perpendicular AB on the time axis (x – axis).

AB = Distance travelled by the body

OB = Time interval

Now, Velocity =

Velocity =

Where is the slope of line OA

Hence, we calculate uniform velocity from the displacement – time graph.

The Velocity – Time graph gives the change of velocity with the change in time.

Y – Axis : Velocity

X – Axis : Time

Now, slope of velocity – time graph is gives as

Slope = =

Since, y = v (Velocity)

And x = t (Time)

Now, there are three types of Velocity – Time graph.

a) Velocity is constant – Uniform Motion

This means that velocity remains same for all time.

b) Uniformly Accelerated Motion

This means that velocity changes at constant rate.

Slope = Constant and the graph is a straight line graph

c) Non – Uniformly Accelerated Motion

This means that velocity changes at a variable rate.

Slope = Variable and the graph is as shown below

Velocity – Time Graph can be used to find the following as:

a) Acceleration:

Slope of Velocity – Time Graph = Acceleration

Since, Slope =

So, Acceleration =

And hence, Slope = Acceleration

b) Displacement of a body:

The area under the velocity – time graph gives the displacement of the body.

Displacement = Velocity × Time

Area under the graph = Displacement of Body

For finding the area, we multiply the velocity and time.

c) Distance of a body:

Distance = Speed × Time

Now, Speed is magnitude of velocity.

Hence, as long as velocity is positive,

Area under the curve = Distance

NOTE:

If velocity is negative, then the area under the curve is negative and then the total area gives displacement. So, in that case,

Distance Displacement

Example:

Let the velocity – time graph be as shown below:

Now, Displacement = Area under the curve

=

= 5 m

Whereas, Distance =

= 15 m

For displacement we have considered the area below the time axis as negative whereas for distance, we have considered all areas as positive.

Initial Speed = u = 40 ms^-1

Acceleration = a = - 2 ms^-2 (Retardation, hence negative)

Final Speed = v

Now, s = Distance train travels before stopping

Now, we know from the equations of motion that,

s = 400 m

So, the train covers 400 m before stopping.

Also the length of platform = 400 m.

So, the train stops in time.

Acceleration = a =2.5 ms^-2

Time of Acceleration = t = 4 s

Initial velocity = u = 0 ms^-1

Now, from the equations of motion we know that,

s = 20 m

Hence, the girl travels 20 m in this time.

Train A

Initial Velocity = U_A = 72 kmh^-1 = 20 ms^-1

Time Taken = t = 50s

Acceleration = a_A = 0 ms^-2

Now, from the equations of motion, we know that,

m

Driver is at starting of Train A

Train B

Initial Velocity = U_A = 72 kmh^-1 = 20 ms^-1 Time Taken = t = 50 s

Acceleration = a_B = 1 ms^-2

Now, from the equations of motion, we know that,

m

Guard is at the end of Train B

Now, length of both trains = 400 m + 400 m = 800 m

Now, Original distance between Train A and B is S, which can be obtained as given below:

S = 2250 m – 1000 m – 800 m

= 450 m

Velocity of Car = 18 ms^-1

To express this velocity in kmhr^-1 we need the conversion factor which is calculated as under:

1 km = 1000 m

So, 1 m = km

Now, 1 hr = 60 minutes = 60 × 60 s = 3600 s

So, 1 s = hr

Now, Velocity of car = 18

= 18

= kmh^-1

= 64.8 kmh^-1

Now, = 120 kmh^-1

= ms^-1

= ms^-1

Now, time = 30 s

Hence, Distance = × time

=

= 1000 m

= 1 km

Hence, the distance travelled by the electric engine in 30 s is 1 km.

In Classical Mechanics, we deal with relative motion, that is, we study motion with respect to an observer.

So, first thing that we need to do is to set an observer and study motion with respect to this observer which is at rest.

a) A man sitting in a train is at rest with respect to a man sitting in the train next to him.

b) A man sitting in a train is in motion with respect to a man standing on ground outside the train, because the train is moving with respect to the observer and so is the man sitting in it.

When I return to home from school, then my displacement = 0.

Displacement = Final Position – Initial Position

Now, initially I was at home and finally also I am at home only. Hence, the displacement = 0

When we apply brakes to the car, then the direction of the acceleration is opposite to that of the motion of car.

Let the car is moving in positive x direction. Now, we apply brake to stop the car. Now for stopping the car force must act on it (Newton’s First Law) in opposite direction. Hence acceleration is in negative x direction. We may also call it as retardation.

If we apply same force on a car and a motorcycle, then the motorcycle will produce more acceleration.

We know that, Force = Mass × Acceleration

Now, force is same on both, but

Mass of Car > Mass of Motorcycle

Now, using above equation of force,

F = m × a

Now, F = Constant

So, a =

a

So, acceleration of car < acceleration of motorcycle

Negative acceleration is nothing but retardation.

Acceleration is rate of change of velocity.

But, negative acceleration means that the rate of change of velocity is negative or velocity decreases.

Example:

1) When we apply brakes in a moving car, then negative acceleration acts on it and the car stops.

2) When we throw a ball upwards, then also negative acceleration acts on it. Thus when the ball reaches the highest point, velocity of the ball becomes zero.

Dynamics is the branch of physics and a sub branch of Mechanics. Dynamics is concerned with the motion of material objects and also concerns with the cause of motion. It deals with the factors that affect motion such as force, mass, momentum and energy.

Whereas, Statics deals with objects at rest and Kinematics deals with objects in motion without considering the cause of motion.

Mr. X is right, since he says “the farther you go the faster you go”, which implies, the more distance we travel in a given time the more is our velocity and thus he is right.

Whereas Mr. Y says “the longer you go the faster you go”. Now, longer means more time. Hence he is saying that more time you take, more the velocity you have, which is completely wrong.

Let “s” be the total distance and “a” be the acceleration.

Now, we know that initially the body is at rest.

Now, from the equations of motion, we get,

…….1

…….2

To find the time taken for covering, we put in equation 1 above.

So, …….3

Now, dividing equation 2 by 3, we get,

t = 2 s

Hence, option B is correct.

We know that, Velocity =

Now, Displacement = Final Position – Initial position

Since, the car comes back to its starting point, so, displacement = 0

Therefore, Velocity = = 0 kmh^-1

Hence, option C is correct

We know that,

Displacement …….1

Now, Velocity = …….2

From equations 1 and 2, we get,

Velocity …….3

Now, we also know that,

Acceleration = …….4

From equations 3 and 4, we get,

Acceleration time

Since, acceleration is directly proportional with time, so it increases with time.

Hence option A is correct.

Now, from the given velocity – time graph we can calculate the distance by calculating area under the graph.

Distance Travelled = Area under the graph

=

= 16 m

Hence option B is correct

From the diagram,

Circumference =

× Circumference =

=

Now,

Distance = Total path covered by the body

= × Circumference

= =

Whereas,

Displacement = Shortest Distance (Straight line) AB

=

=

= r

Hence, option B is correct.

Now, Distance = Total Path Covered

And Displacement = Shortest distance between initial and final point

So, Clearly Distance Displacement

Or, Displacement Distance

So,

But Distance = Displacement is valid only on a straight line path where direction does not change.

Hence option A is correct.

Velocity at A = u = 20 ms^-1

Velocity at B = v = 30 ms^-1

Acceleration = a

Let AB = x

Now, from the equations of motion, we know that,

a =

Now, at mid – point M, Velocity = v_m

Also, AM =

So,

ms^-1

Hence, option C is correct.

We know that, Distance …..1

Now, Velocity = …..2

From equations 1 and 2, we get,

Velocity Time …..3

Now, we also know that, Acceleration = …..4

So, from equations 3 and 4, we get,

Acceleration

Acceleration = Constant

Hence, option D is correct

Area under the Velocity – Time graph gives the displacement.

Now, Displacement = Velocity × time

Also, area under the graph is calculated by multiplying velocity and time. Thus displacement is obtained in that process. Hence option A is correct.

Speed is not a vector quantity. Speed is distance travelled per unit time. It has only magnitude and no direction. Whereas for velocity, displacement and acceleration, all have magnitude as well as direction.

Hence option C is correct.

If the velocity of the body is always changing with a uniform rate (uniform acceleration), then the average velocity is given by,

Average Velocity =

Hence option D is correct.

If speed – time graph is parallel to time axis, then this means that the body has same speed at all time.

Thus, Speed = Constant

Hence the body has uniform speed and thus option B is correct.

The graph says that the velocity is decreasing with time. Now, slope of velocity – time graph gives acceleration.

Hence acceleration is the rate of change of velocity. Since velocity is decreasing, hence acceleration is also negative. Now, negative acceleration is also called retardation.

Since the slope is constant, hence the body has uniform retardation. Hence option B is correct.

The body does not move as the body is at same position at all times.

Thus body is at rest.

Hence option A is correct.

Slope of the distance – time graph gives speed.

Here in the adjoining graph, slope is constant.

Now, since slope is constant, then it means that the body is moving with constant velocity or speed. Hence body is in uniform motion and thus option A is correct.

We know that,

Displacement = Velocity × Time

So, the area under the curve of Velocity – Time graph of a body will give the magnitude of displacement of the body. Hence option C is correct.

We know that,

Distance = Speed × Time

So, area under the Speed – Time graph will give the distance covered by the particle. Hence the option C is correct.

The direction of acceleration of an object having a circular path is directed towards the center of circle.

This acceleration is called the centripetal acceleration. This centripetal acceleration produces centripetal force to change the direction of the object, so that it moves on a circular path.

In the above figure: a_c = Centripetal Acceleration and v = Speed of Object.

Hence option B is correct.

True

Mechanics is a branch of physics that deals with motion of non – living objects.

The motion of animals is locomotion and the laws of mechanics cannot be applied to it. Locomotion is a complex phenomenon of movement of different body parts and transition from one place to another.

True

Kinematics is a branch of physics that deals with motion without taking into consideration the cause of motion.it deals with only position, velocity and acceleration. It does not talk about the cause of motion, like Force.

False

Motion along a curved path is not called translator or rectilinear motion.

Motion along a curved path is called a curvilinear motion. A curvilinear motion is accelerated motion as direction changes.

True

Whenever we talk about the motion of a body, we talk with respect to a reference frame that is at rest. So, if particles’ position does not change with time, which means that the velocity of the particle is zero, then the body can be considered at rest. This can also be depicted via the distance – time graph, where velocity = = 0.

False

A quantity which can be represented completely by magnitude alone is not a vector quantity, but a scalar quantity.

A scalar quantity has only magnitude.

Example can be distance, speed, etc.

False

A quantity which can be completely specified by magnitude as well as direction is not a scalar quantity, but a vector quantity.

A vector quantity has both magnitude and direction.

Examples can be displacement, velocity, etc.

False

Velocity and Speed are not measured in different units. They both have same unit that is ms^-1.

Velocity =

Speed =

Thus from above we can clearly infer that both velocity and speed have same units of measurement.

False

In one dimensional motion average velocity and instantaneous velocity are not always unequal.

They are equal when motion of the body is a uniform motion that is if the velocity remains constant.

True

Velocity =

Now, equal displacement in equal interval of time means = Constant

Thus, Velocity = Constant and hence the motion is uniform motion.

False

A motion is said to be uniform if velocity is constant.

Velocity is constant means velocity does not change with time.

Now, we know that,

Displacement = x …..1

Also, we know that,

Velocity = …..2

From equations 1 and 2, we get,

Velocity Time

So, velocity increases with time and hence not constant. So, if x , then motion is not uniform.

True

Yes, acceleration is defined as the rate of change of velocity.

Acceleration =

From the V – T Graph shown here, we get slope of the graph as:

Slope =

Slope = Acceleration

False

Slope of Velocity – Time graph gives acceleration.

Acceleration = = Slope

Now, for uniform acceleration, acceleration should be constant, which would imply that the slope will be constant and hence the velocity – time curve should be straight line as shown in the above figure.