A roller coaster car travels down the helical path at constant speed such that it's parametric coordinates varies as `x = c sin(kt), y = c cos(kt), z = h - bt` where c, h, k and b are constants, then the megnitude of it's acceleration is :

Options

A. `0`
B. `ck^(2)`
C. `(c^(2)k^(2))/(h)`
D. `bk^(2)`

Explanation

To solve the problem, we need to find the magnitude of the acceleration of a roller coaster car moving along a helical path defined by the parametric equations:

- \( x = c \sin(kt) \)
- \( y = c \cos(kt) \)
- \( z = h - bt \)

where \( c, h, k, \) and \( b \) are constants.

### Step 1: Find the velocity components

1. **Differentiate \( x \) with respect to \( t \)**:
\[ V_x = \frac{dx}{dt} = \frac{d}{dt}(c \sin(kt)) = c k \cos(kt) \]

2. **Differentiate \( y \) with respect to \( t \)**:
\[ V_y = \frac{dy}{dt} = \frac{d}{dt}(c \cos(kt)) = -c k \sin(kt) \]

3. **Differentiate \( z \) with respect to \( t \)**:
\[ V_z = \frac{dz}{dt} = \frac{d}{dt}(h - bt) = -b \]

### Step 2: Find the acceleration components

1. **Differentiate \( V_x \) with respect to \( t \)**:
\[ A_x = \frac{dV_x}{dt} = \frac{d}{dt}(c k \cos(kt)) = -c k^2 \sin(kt) \]

2. **Differentiate \( V_y \) with respect to \( t \)**:
\[ A_y = \frac{dV_y}{dt} = \frac{d}{dt}(-c k \sin(kt)) = -c k^2 \cos(kt) \]

3. **Differentiate \( V_z \) with respect to \( t \)**:
\[ A_z = \frac{dV_z}{dt} = 0 \]

### Step 3: Calculate the magnitude of the acceleration

The magnitude of the acceleration \( A \) is given by:
\[ A = \sqrt{A_x^2 + A_y^2 + A_z^2} \]

Substituting the values we found:
\[ A = \sqrt{(-c k^2 \sin(kt))^2 + (-c k^2 \cos(kt))^2 + 0^2} \]
\[ = \sqrt{c^2 k^4 \sin^2(kt) + c^2 k^4 \cos^2(kt)} \]
\[ = \sqrt{c^2 k^4 (\sin^2(kt) + \cos^2(kt))} \]
Using the identity \( \sin^2(kt) + \cos^2(kt) = 1 \):
\[ = \sqrt{c^2 k^4} = c k^2 \]

### Final Result
Thus, the magnitude of the acceleration is:
\[ A = c k^2 \]