The equation for the change of position of a train starting at x

= 0 is given by
x=1/2at^2 + bt3 . Find the dimensions of a and b.

If y = C1 sin (C2 t)
where y is a distance and t is the time. What are the dimensions of C1 and
C2 ?.

To find the dimensions of a and b in the equation x = (1/2)at^2 + bt^3, we can analyze the equation and compare the units on both sides.

The equation x = (1/2)at^2 + bt^3 represents the change of position of a train starting at x = 0. Let's break down the terms in the equation:

1. The term (1/2)at^2 represents the displacement due to constant acceleration. The dimensions of (1/2)at^2 would be [L] (length) since a is acceleration (L/T^2) and t is time (T).

2. The term bt^3 represents the displacement due to cubic term in time. The dimensions of bt^3 would be [L] (length) since b is a constant and t is time (T).

Since both terms on the right side of the equation represent a distance, the dimensions of a and b should both be [LT^-2] for a and [LT^-3] for b.

Now let's consider the second equation y = C1 sin (C2 t), where y is a distance and t is time.

1. The term C1 represents the amplitude or maximum displacement of the sine wave. The dimensions of C1 would be [L] (length) since it represents a distance.

2. The term C2 represents the angular frequency or how fast the waveform cycles. The dimensions of C2 would be [T^-1] (time inverse) since it represents cycles per unit time.

Therefore, the dimensions of C1 would be [L] (length) and the dimensions of C2 would be [T^-1] (time inverse).

x=1/2 a t^2 + bt^3

if at^2 is distance, then a must be distance/time^2

b has to be distance per time^3

If y is distance, C1 must be distance. C2 must be 1/time

if at^2 is distance, then a must be distance/time^2

okay but from where can i get distance ?!
i mean as a number !

also here sin (C2 t) i think it is constant so we can't take dimension or i am wrong :(

Again Thank u a lot :D

You are solving dimensional analysis here, just the dimensions. There are no numbers.

The argument in a trig function is in radians, So C2*time must divide out all units, so C2 is 1/time.

should we divied it , I mean solve it like

x = 1/2at^2 ,x= bt^3 and take each part individual !

how can get red of (sin) from this formula ,
also here should i handle it individual !