
If the displacement of a body is proportional to the square of time, then the body is moving with
A. Uniform acceleration
B. Increasing acceleration
C. Decreasing acceleration
D. Uniform velocity
Answer
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Hint:Write the equation for the displacement of the body as a function of time. Then differentiate the displacement with respect to time and find the velocity. After this, differentiate the velocity with respect to time to find the acceleration.
Formula used:
Complete step by step answer:
It is given that the displacement of a body is proportional to the square of time.
Therefore, we can write that …. (i),
where x is the displacement of the body and t is time.
This means that if the time is increased by factor ‘n’ then the displacement of the particle will increase by a factor of .
By adding a proportionality constant k to (i), we can write an equation for x as ….. (ii).
We can see that the options are stating about the velocity and the acceleration of a body.
Velocity (v) of a body is the first derivative of displacement with respect to time.
Therefore, differentiate (ii) with respect to time t.
…. (iii)
Now, we can see that the velocity of the given body is directly proportional to t.
Acceleration (a) of a body is the first derivative of its velocity with respect to time t.
Therefore, differentiate (iii) with respect to time t.
.
We know that k is a constant. Therefore, the acceleration of the body is constant.
Constant acceleration is also called uniform acceleration.
Hence, the correct option is A.
Note:If you are well known with the kinematic equations for uniform acceleration, then this would be a very easy problem.
One of the kinematic equations say that , where u is the initial velocity of the body.
If we put a condition that the body was at rest initially ( ), then the displacement of the body is equal to .
.
Hence, the body is travelling with uniform acceleration.
Formula used:
Complete step by step answer:
It is given that the displacement of a body is proportional to the square of time.
Therefore, we can write that
where x is the displacement of the body and t is time.
This means that if the time is increased by factor ‘n’ then the displacement of the particle will increase by a factor of
By adding a proportionality constant k to (i), we can write an equation for x as
We can see that the options are stating about the velocity and the acceleration of a body.
Velocity (v) of a body is the first derivative of displacement with respect to time.
Therefore, differentiate (ii) with respect to time t.
Now, we can see that the velocity of the given body is directly proportional to t.
Acceleration (a) of a body is the first derivative of its velocity with respect to time t.
Therefore, differentiate (iii) with respect to time t.
We know that k is a constant. Therefore, the acceleration of the body is constant.
Constant acceleration is also called uniform acceleration.
Hence, the correct option is A.
Note:If you are well known with the kinematic equations for uniform acceleration, then this would be a very easy problem.
One of the kinematic equations say that
If we put a condition that the body was at rest initially (
Hence, the body is travelling with uniform acceleration.
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