Motion of an object

We see the motion of several objects every day. Sometimes we cannot see the motion of an object directly, as in the case of a breeze. Can you list other examples of motion, besides those given here?

Displacement and distance

‘Distance’ is the length of the actual path travelled by an object in motion while going from one point to another, whereas displacement is the minimum distance between the starting and finishing points.

Speed and velocity

The distance travelled in one direction by an object in unit time is called its velocity. Here, unit time can be one second, one minute, one hour, etc. If large units are used, one year can also be used as a unit of time. The displacement that occurs in unit time is called velocity.

In the first example (on page 1), the straight line distance between the houses of Sheetal and Sangeeta is 500 m and that between Sangeeta’s house and school is 1200 m. Also, the straight line distance between Sheetal’s house and school is 1300 m. Suppose Sheetal takes 5 minutes to reach Sangeeta’s house and then 24 minutes to reach school from there, Then,

Effect of speed and direction on velocity

Sachin is travelling on a motorbike. Explain what will happen in the following events during Sachin’s ride (see figure 1.3).

1. What will be the effect on the velocity of the motorcycle if its speed increases or decreases, but its direction remains unchanged?
2. In case of a turning on the road, will the velocity and speed be same? If Sachin changes the direction of the motorcycle, keeping its speed constant, what will be the effect on the velocity?
3. If, on a turning, Sachin changes the direction as well as the speed of the motorcycle, what will be the effect on its velocity? It is clear from the above that velocity depends on speed as well as direction and that velocity changes by
4. changing the speed while keeping the direction same
5. changing the direction while keeping the speed same
6. changing the speed as well as the direction

Uniform and non-uniform linear motion

Amar, Akbar and Anthony are travelling in different cars with different velocities. The distances covered by them during different time intervals are given in the following table

Acceleration

1. Take a 1m long plastic tube and cut it lengthwise into two halves.
2. Take one of the channel shaped pieces. Place one of its ends on the ground and hold the other at some height from the ground as shown in figure 1.4.
3. Take a small ball and release it from the upper end of the channel.
4. Observe the velocity of the ball as it rolls down along the channel.
5. Is its velocity the same at all points?
6. Observe how the velocity changes as it moves from the top, through the middle and to the bottom.

You must have all played on a slide in a park. You know that while sliding down, the velocity is less at the top, it increases in the middle and becomes zero towards the end. The rate of change of velocity is called acceleration

If the velocity of an object changes during a certain time period, then it is said to have accelerated motion. An object in motion can have two types of acceleration.

1. When an object is at rest in the beginning of its motion, Change in velocity Time. Final velocity – Initial velocity Time (v-u) t Acceleration = a = \ a = what is its initial velocity?
2. When an object comes to rest at the end of its motion, what is its final velocity?

Positive, negative and zero acceleration

An object can have positive or negative acceleration. When the velocity of an object increases, the acceleration is positive. In this case, the acceleration is in the direction of velocity. When the velocity of an object decreases with time, it has negative acceleration. Negative acceleration is also called deceleration. Its direction is opposite to the direction of velocity. If the velocity of the object does not change with time, it has zero acceleration.

Distance-time graph for uniform motion

 The following table shows the distances covered by a car in fixed time intervals. Draw a graph of distance against time taking ‘time’ along the X-axis and ‘distance’ along the Y-axis in figure 1.5.

An object in uniform motion covers equal distances in equal time intervals. Thus, the graph between distance and time is a straight line.

Distance-time graph for non-uniform motion

The following table shows the distances covered by a bus in equal time intervals Draw a graph of distance against time taking the time along the X-axis and distance along the Y-axis in figure 1.6. Does the graph show a direct proportionality between distance and time?

Velocity-time graph for uniform velocity

A train is moving with a uniform velocity of 60 km/hour for 5 hours. The velocity-time graph for this uniform motion is shown in figure 1.7.

1. With the help of the graph, how will you determine the distance covered by the train between 2 and 4 hours?
2. Is there a relation between the distance covered by the train between 2 and 4 hours and the area of a particular quadrangle in the graph? What is the acceleration of the train?

Velocity-time graph for uniform acceleration

The changes in the velocity of a car in specific time intervals are given in the following table.

The velocity-time graph in figure 1.8 shows that, 1. The velocity changes by equal amounts in equal time intervals. Thus, this is uniform acceleration in accelerated motion. How much does the velocity change every 5 minutes? 2. For all uniformly accelerated motions, the velocity-time graph is a straight line. 3. For non-uniformly accelerated motions, the velocity-time graph may have any shape depending on how the acceleration changes with time. From the graph in figure 1.8, we can determine the distance covered by the car between the 10th and the 20th seconds as we did in the case of the train in the previous example. The difference is that the velocity of the car is not constant (unlike that of the train) but is continuously changing because of uniform acceleration. In such a case, we have to use the average velocity of the car in the given time interval to determine the distance covered in that interval.

From the graph, the average velocity of the car = 32+16 /2 = 24 m/s

Multiplying this by the time interval, i.e. 10 seconds gives us the distance covered by the car. Distance covered = 24 m/s x 10 s = 240 m Check that, similar to the example of the train, the distance covered is given by the area of quadrangle ABCD.

A ( ABCD ) = A ( AECD ) + A ( ABE )

Equations of motion using graphical method

Newton studied motion of an object and gave a set of three equations of motion. These relate the displacement, velocity, acceleration and time of an object moving along a straight line.

Suppose an object is in motion along a straight line with initial velocity ‘u’. It attains a final velocity ‘v’ in time ‘t’ due to acceleration ‘a’ its desplacement is ‘s’. The three equations of motion can be written as

Equation describing the relation between velocity and time

Figure 1.9 shows the change in velocity with time of a uniformly accelerated object. The object starts from the point D in the graph with velocity u. Its velocity keeps increasing and after time t, it reaches the point B on the graph.

The initial velocity of the object = u = OD The final velocity of the object = v = OC Time = t = OE

Draw a line parallel to Y axis passing through B. This will cross the X axis in E. Draw a line parallel to X-axis passing through D. This will cross the line BE at A.

From the graph…. BE = AB + AE

 v = CD + OD …..(AB = CD and AE = OD)

 v = at + u …………(from i )

 v = u + at This is the first equation of motion.

Equation describing the relation between displacement and time

Let us suppose that an object in uniform acceleration ‘a’ and it has covered the distance ‘s’ within time ‘t’. From the graph in figure 1.9, the distance covered by the object during time ‘t’ is given by the area of quadrangle DOEB.

 s = area of quadrangle DOEB = area (rectangle DOEA) + area of triangle (DAB) \

s = (AE × OE ) + ( × [AB × DA])

But, AE = u, OE = t and (OE = DA = t) AB = at —( AB = CD ) — from (i)

Equation describing the relation between displacement and velocity

We have seen that from the graph in figure 1.9 we can determine the distance covered by the object in time t from the area of the quadrangle DOEB. DOEB is a trapezium. So we can use the formula for its area.  s = area of trapezium DOEB

Uniform circular motion

The speed of the tip of a clock is constant, but the direction of its displacement and therefore, its velocity is constantly changing. As the tip is moving along a circular path, its motion is called uniform circular motion. Can you give more examples of such motion?

When an object is moving with a constant speed along a circular path, the change in velocity is only due to the change in direction. Hence, it is accelerated motion. When an object moves with constant speed along a circular path, the motion is called uniform circular motion, e.g. the motion of a stone in a sling or that of any point on a bicycle wheel when they are in uniform motion.

Determining the direction of velocity in uniform circular motion.

Take a circular disc and put a five rupee coin at a point along its edge.

Make it move around its axis by putting a pin through it. When the disc is moved at higher speed, the coin will be thrown off as shown in figure 1.11. Note the direction in which it is thrown off. Repeat the action placing the coin at different points along the edge of the circle and observe the direction in which the coin is thrown off.

The coin will be thrown off in the direction of the tangent which is perpendicular to the radius of disc. Thus, the direction in which it gets thrown off depends on its position at the moment of getting thrown off. It means that, as the coin moves along a circular path the direction of its motion is changing at every point.

Newton’s laws of motion

What could be the reason for the following?

1. A static object does not move without the application of a force.
2. The force which is sufficient to lift a book from a table is not sufficient to lift the table.
3. Fruits on a tree fall down when its branches are shaken.
4. An electric fan keeps on rotating for some time even after it is switched off.

If we look for reasons for the above, we realize that objects have some inertia. We have learnt that inertia is related to the mass of the object. Newton’s first law of motion describes this very property and is therefore also called the law of inertia.

Newton’s first law of motion

Balanced and unbalanced force

You must have played tug-of-war. So long as the forces applied by both the sides are equal, i.e. balanced, the centre of the rope is static in spite of the applied forces. On the other hand, when the applied forces become unequal, i.e. unbalanced, a net force gets applied in the direction of the greater force and the centre of the rope shifts in that direction.

 ‘An object continues to remain at rest or in a state of uniform motion along a straight line unless an external unbalanced force acts on it.’

When an object is at rest or in uniform motion along a straight line, it does not mean that no force is acting on it. Actually there are a number of forces acting on it, but they cancel one another so that the net force is zero. Newton’s first law explains the phenomenon of inertia, i.e. the inability of an object to change its state of motion on its own. It also explains the unbalanced forces which cause a change in the state of an object at rest or in uniform motion.

All instances of inertia are examples of Newton’s first law of Motion.

Newton’s second law of motion

The effect of one object striking another object depends both on the mass of the former object and its velocity. This means that the effect of the force depends on a property related to both mass and velocity of the striking object. This property was termed ‘momentum’ by Newton.

Momentum (P) : Momentum is the product of mass and velocity of an object. P = m v. Momentum is a vector quantity.

‘The rate of change of momentum is proportional to the applied force and the change of momentum occurs in the direction of the force.’

Suppose an object of mass m has an initial velocity u. When a force F is applied in the direction of its velocity for time t, its velocity becomes v.

 The initial momentum of the object = mu,

Its final momentum after time t = mv

Consider two objects having different masses which are initially at rest. The initial momentum for both is zero. Suppose a force ‘F’ acts for time ‘t’ on both objects. The lighter object starts moving faster than the heavier object. However, from the above formula, we know that the rate of change of momentum i.e. ‘F’ in both objects is same and the total change in their momentum will also be same i.e. ‘Ft’. Thus, if the same force is applied on different objects, the change in momentum is the same.

Newton’s third law of motion

We have learnt about force and its effect on an object through Newton’s first and second laws of motion.

 ‘However, in nature force cannot act alone.’ Force is a reciprocal action between two objects. Forces are always applied in pairs. When one object applies a force on another object, the latter object also simultaneously applies a force on the former object. The forces between two objects are always equal and opposite. This idea is expressed in Newton’s third law of motion. The force applied by the first object is called action force while the force applied by the second object on the first is called reaction force.

 ‘Every action force has an equal and opposite reaction force which acts simultaneously.’

Law of conservation of momentum

Suppose an object A has mass m1 and its initial velocity is u1 . An object B has mass m2 and initial velocity u2 .

 According to the formula for momentum, the initial momentum of A is m1 u1 and that of B is m2 u2 .

Suppose these two objects collide. Let the force on A due to B be F1 . This force will cause acceleration in A and its velocity will become v1 . \

Momentum of A after collision = m1 v1 According to Newton’s third law of motion, A also exerts an equal force on B but in the opposite direction. This will cause a change in the momentum of B. If its velocity after collision is v2 , The momentum of B after collision = m 2 v 2 . If F 2 is the force that acts on object B,

The magnitude of total final momentum = the magnitude of total initial momentum.

Thus, if no external force is acting on two objects, then their total initial momentum is equal to their total final momentum. This statement is true for any number of objects.

 ‘When no external force acts on two interacting objects, their total momentum remains constant. It does not change.’

This is a corollary to Newton’s third law of motion. The momentum is unchanged after the collision. The momentum gets redistributed between the colliding objects. The momentum of one of the objects decreases while that of the other increases. Thus, we can also state this corollary as follows.

‘When two objects collide, the total momentum before collision is equal to the total momentum after collision.’

 In order to understand this principle, let us consider the example of a bullet fired from a gun. When a bullet of mass m1 is fired from a gun of mass m2 , its velocity becomes v1 , and its momentum becomes m1 v1 . Before firing the bullet, both the gun and the bullet are at rest and hence the total initial momentum is zero. According to the above law, the total final momentum also has to be zero. Thus, the forward moving bullet causes the gun to move backward after firing. This backward motion of the gun is called its recoil. The velocity of recoil, v2 , is such that,