Motion - Olympiad Level MCQ, Class 9 Science - Class 9 MCQ

Let the total distance be 3x km. Now, average speed is given by 

average speed = (total distance travelled) / (total time taken)

So, let times taken for the first, second and third halves of the journeys be t1, t2, and t3 respectively. Then, 

t1 = x/10

t2 = x/20

t3 = x/60

So, average speed = x / (t1 + t2 + t3)

= x / (x/10 + x/20 + x/60)

= 3/ ((6 + 3 + 1)/60)

= (3*60)/10

= 18 km/hr

The correct option is A.

speed = distance / time

or

time = distance / speed

here,

distance between earth and moon = 4 x 10⁸ m

speed of radar signal = 3 x 10⁸m/s [speed of light ]

so,

the time taken by radar signal to reach the moon will be

t =  4 x 10⁸ / 3 x 10⁸

thus,

t = 1.33 s

Given:
- Weight of the stone (m) = 3 kg
- Height of the tower (h) = 100 m
- Depth of the penetration in the sand (d) = 2 m
We need to calculate the time of penetration.
Step 1: Calculating the Initial Velocity (u)
To calculate the initial velocity, we can use the equation:
v^2 = u^2 + 2gh
where,
v = final velocity (which is 0 as the stone comes to rest)
u = initial velocity
g = acceleration due to gravity (which is approximately 9.8 m/s^2)
h = height of the tower
Substituting the values, we have:
0 = u^2 + 2 * 9.8 * 100
0 = u^2 + 1960
u^2 = -1960
Since velocity cannot be negative, we discard the negative value. Therefore,
u = √1960 ≈ 44.27 m/s
Step 2: Calculating the Time of Penetration (t)
We can use the equation of motion:
s = ut + (1/2)at^2
where,
s = distance traveled (which is the sum of the height of the tower and the depth of penetration)
u = initial velocity
t = time of penetration
a = acceleration due to gravity (which is approximately 9.8 m/s^2)
Substituting the values, we have:
100 + 2 = 44.27 * t + (1/2) * 9.8 * t^2
102 = 44.27t + 4.9t^2
4.9t^2 + 44.27t - 102 = 0
Solving this quadratic equation, we find that:
t ≈ 0.0903 sec or t ≈ -2.703 sec
Since time cannot be negative, we discard the negative value. Therefore,
t ≈ 0.0903 sec
Hence, the time of penetration is approximately 0.09 seconds.
Therefore, the correct answer is option A: 0.09 sec.

Given information:
- The velocity of a body at any instant is 10 m/s.
- After 5 seconds, the velocity of the particle is 20 m/s.
To find:
- The velocity at 3 seconds before.

Let's assume the initial velocity of the body as u and the final velocity as v.
From the given information, we have:
- Initial velocity, u = 10 m/s
- Final velocity, v = 20 m/s
- Time, t = 5 seconds
We can use the formula for velocity:
v = u + at
where a is the acceleration. Since the problem does not provide any information about acceleration, we can assume it to be constant.
Substituting the given values, we have:
20 = 10 + a * 5
Simplifying the equation, we get:
10 = 5a
a = 2 m/s²
Now, we can find the velocity at 3 seconds before using the formula:
v = u + at
where t is the time before the given time.
Substituting the values:
- Initial velocity, u = 10 m/s
- Time, t = 5 - 3 = 2 seconds
- Acceleration, a = 2 m/s²
v = 10 + 2 * 2
v = 10 + 4
Therefore, the velocity at 3 seconds before is 14 m/s. Hence, option B is correct.

Let 'u' be the initial velocity and 'a' the acceleration.

So we have the distance formula
s = ut + 1/2 at^2

In first 2 seconds,
s = 200
Put the values in the formula.
200 = u x 2 + 1/2 x a x (2)^2
200 = 2u + 1/2 x 4 x a
200 = 2u + 2a
100 = u + a -----(I)

In next 4 seconds
Let the distance traveled be 'y'
Time = 2+4 = 6 sec
So according to the formula ,
y = u x 6 + 1/2a x (6)^2
y = 6u + 1/2 x 36 x a
y = 6u + 18a ------(ii)

Now we know that 220 cm was traveled in between 2 sec - 6 sec
y - 200 = 220
y = 420

We know y = 6u + 18a (from [ii])

So, 6u + 18a = 420
u + 3a = 70 ------(iii)

Equating (I) and (iii)
-2a = 30
a = -15 cm/s^2

u = 100 - (-15)
u = 100 + 15 = 115 cm/sec

Now we know v = u + at

We have to find the "v" after 7th second

So v = 115 + (-15) x 7
v = 115 - 105
v = 10 cm/sec

For a freely falling object :
a=g;

Initial velocity=u=0 m/s

g=9.8m/s²

from second equation of motion:S=ut+1/2 at²

for First object we get :

h1= gT1²/2

⇒T1=√2h1/g

For second object we get h2=gT2²/2

⇒T2=√2h2/g

T1/T2 =√h1/h2

But here h1= a and h2 =b
so, T1/T2=√a/b

H = (v² - u²) / (2g)
H = (9u² - u²) / (2g)
H = 8u² / (2g)
H = 4u²/g
Height of the tower is 4u²/g

Since the ball was thrown from a height and returns to the same height therefore initial position = final position. since displacement = Straight line distance between initial and final position therefore displacement = 0. since the ball attains the height of 40 m therefore while returningit covers the same distance.therefore distance = 2× 40 m = 80 m.

Acceleration of a body projected upwards with a certain velocity is
To determine the acceleration of a body projected upwards with a certain velocity, we need to consider the forces acting on the body and apply Newton's second law of motion.
Newton's Second Law of Motion:
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It can be stated as:
F = ma
Where:
F is the net force acting on the object,
m is the mass of the object,
a is the acceleration of the object.
For a body projected upwards, the following forces are acting:
1. Gravitational force (weight) acting downwards.
2. Air resistance (if present) acting upwards.
Analysis:
1. When a body is projected upwards, the initial velocity is in the upward direction.
2. The only force acting on the body is the gravitational force (weight) acting downwards.
3. The gravitational force is always directed towards the center of the Earth and is equal to the mass of the body multiplied by the acceleration due to gravity (9.8 m/s^2).
4. As the body moves upwards, the gravitational force opposes its motion, causing it to decelerate.
5. Therefore, the acceleration of the body projected upwards is negative (-9.8 m/s^2).
Conclusion:
The acceleration of a body projected upwards with a certain velocity is -9.8 m/s^2 (Option B).

Linear velocity V = angular velocity × radius. Angular velocity = [30×2π]/60 R= 14s. So V = 44 cm/s.

Relationship between average speed, time, and distance:

• Average speed = Distance / Time: This is the correct relationship between average speed, distance, and time. It states that the average speed is calculated by dividing the total distance traveled by the time taken to cover that distance.


• Time = Distance / Average speed: This is not the correct relationship. Time cannot be determined by dividing distance by average speed. Instead, it is obtained by dividing distance by average speed.


• Distance = Average speed × Time: This is not the correct relationship. Distance cannot be determined by multiplying average speed by time. Instead, it is obtained by multiplying average speed by time.

Therefore, the correct relationship between average speed, time, and distance is:
Average speed = Distance / Time (Option C).

Explanation:

When a body moves along a circular path, it experiences certain characteristics:

• Constant speed: The body moves at a constant rate around the circle, covering equal distances in equal intervals of time.


• Constant velocity: The body's velocity, which includes both speed and direction, remains constant as long as it moves at a uniform rate along the circular path.


• No radial acceleration: The body does not experience any radial acceleration because the distance from the center of the circle remains constant.


• No tangential velocity: The body's tangential velocity, which is the velocity along the tangent of the circular path, changes as it moves along the circle, but it does not become zero.

Based on these characteristics, we can conclude that a body moving along a circular path has constant speed. Therefore, option A is correct.

Answer:
Given: The velocity of a body does not change.
To find: The acceleration of the body.

When the velocity of a body does not change, it means that the body is moving with a constant velocity. In this case, the acceleration of the body is zero.
Explanation:
Acceleration is defined as the rate of change of velocity. If the velocity of a body does not change, it means there is no change in the rate at which the body is moving. Therefore, the acceleration of the body is zero.
Conclusion:
If the velocity of a body does not change, its acceleration is zero.

Explanation:
When the distance an object travels is directly proportional to the length of time, it means that for every unit of time, the object travels a constant distance. This indicates that the object is moving with a constant speed. Here's a detailed explanation:
Direct Proportion:
In a direct proportion, two quantities are related in such a way that an increase or decrease in one quantity results in a corresponding increase or decrease in the other quantity. In this case, the distance traveled by the object is directly proportional to the time taken.
Constant Speed:
When an object travels with a constant speed, it means that it covers equal distances in equal intervals of time. In other words, the object maintains a steady rate of motion, covering the same amount of distance for each unit of time.
Example:
To illustrate this concept, let's consider a car traveling on a straight road. If the car travels at a constant speed of 60 kilometers per hour, it will cover a distance of 60 kilometers in one hour, 120 kilometers in two hours, and so on. The distance traveled by the car is directly proportional to the time elapsed.
Conclusion:
Therefore, when the distance an object travels is directly proportional to the length of time, it indicates that the object is traveling with a constant speed. So, the correct answer is option B: constant speed.

Analysis:
To solve this problem, we need to understand the concepts of displacement and distance.
Displacement:
Displacement is a vector quantity that represents the change in position of an object. It is the straight-line distance between the initial and final positions of the object, along with its direction.
Distance:
Distance is a scalar quantity that represents the total path length traveled by an object. It does not take into account the direction of motion, only the magnitude.

Given that the body moves on three quarters of a circle of radius r, we can analyze the displacement and distance traveled as follows:
Displacement:
The body moves on three quarters of a circle, which means it starts and ends at the same point. Therefore, the displacement is zero.
Distance:
To calculate the distance traveled, we need to find the length of the three-quarters of the circumference of the circle.
- The circumference of a circle is given by 2πr.
- Three-quarters of the circumference is equal to (3/4) * 2πr = (3π/2)r.
Therefore, the distance traveled by the body is (3π/2)r.
Conclusion:
Based on the analysis, we can conclude that the correct answer is option B: displacement = 0 and distance = (3π/2)r.

To solve this problem, we need to understand the definitions of displacement and distance covered in the context of motion with constant acceleration.
- Displacement: Displacement refers to the change in position of an object. It is a vector quantity and is measured in units of length (meters, kilometers, etc.). Displacement takes into account both the magnitude (distance) and direction of the change in position.
- Distance Covered: Distance covered refers to the total length of the path traveled by an object. It is a scalar quantity and is measured in units of length. Distance covered only considers the magnitude of the path traveled and does not take into account the direction.
Now, let's analyze the given options and determine the correct ratio:
A: = 1
This option states that the ratio of displacement to distance covered is equal to 1. This means that the magnitude of displacement and the distance covered are the same. Since displacement is a vector quantity and takes into account both magnitude and direction, this option is correct.
B:
This option suggests that the ratio of displacement to distance covered is greater than 1. However, this is not possible as displacement cannot be greater than the total distance covered.
C:
This option suggests that the ratio of displacement to distance covered is less than 1. However, this is also not possible as displacement cannot be less than the total distance covered.
D: < 1
This option suggests that the ratio of displacement to distance covered is less than 1, which is not possible.
Therefore, the correct answer is option A: = 1.