Question

A particle moving with a uniform acceleration travels 24 m and 64 m in the first two consecutive intervals of 4 s each. Its initial velocity of the particle is?
A. 5 \[m{s^{ - 1}}\]
B. 3 \[m{s^{ - 1}}\]
C. 4 \[m{s^{ - 1}}\]
D. 1 \[m{s^{ - 1}}\]

Answer 1

Hint: The most basic principles of an object's motion are described by the equations of motion used in kinematics. These equations control how an object moves in the first, second, and third dimensions. They make it simple to compute expressions like an object's position, velocity, or acceleration over time. Sir Isaac Newton identified three laws of motion that control how all the objects in our environment move.

Formula used:
\[s{\text{ }} = {\text{ }}ut{\text{ }} + \dfrac{1}{2}a{t^2}\]
\[Distance = \text{Initial Velocity} \times \text{Time} + \left( {\dfrac{1}{2}} \right) \text{acceleration} \times \text{time}^2\]

Complete step by step solution:
The most basic principles of an object's motion are described by the equations of motion used in kinematics. These equations control how an object moves in the first, second, and third dimensions. They make it simple to compute expressions like an object's position, velocity, or acceleration over time. Sir Isaac Newton identified three laws of motion that control how all the objects in our environment move. The second law of motion is denoted by:
\[s = ut + \left( {\dfrac{1}{2}} \right)a{t^2}\]

According to the question, in the first case, the distance traveled by the particle, s = 24 m and the time taken to cover this distance is, t = 4s.
Putting these given values in the second equation of motion, we get \[24 = 4u + \dfrac{{a{{\left( 4 \right)}^2}}}{2}\]
⇒ 24 = 4u+8a
⇒ 6 = u+2a …………………… equation 1
In the second case, the body travels a total distance of 24 + 64 = 88 meters in total time period of, t = 8s,

Once again putting these values in the equation of motion, we get:
 \[88 = 8u + \dfrac{{a{{\left( 8 \right)}^2}}}{2}\]
⇒ 88 = 8u+32a
⇒ 11 = u+4a …………………..equation 2
Solving the equations. 1 and 2, we get:
\[u = 1\,m{s^{ - 1}}\]

Hence, the correct answer is option (D).

Note: Be careful while solving the equations and always change the units given in the question to suitable ones. Initial velocity is simply the velocity at which the body is started to be observed. There should be no confusion in differentiating an object's initial and final velocity. The initial velocity is the velocity measurement when the object starts to be observed while the final velocity is the velocity at the point of time when the observation of the body is stopped.