Question

The car passes point A with a speed of 25 m/s after which its speed is defined by v = (25-0.15s) m/s.

Determine the magnitude of the car's acceleration when it reaches point B, where s = 51.5 m and x = 50 m. Express your answer to three significant figures and include the appropriate units.

Answer 1

Concepts and reason

The problem deals with the concept of the radial acceleration and translational acceleration as the path of the movement includes both linear and curve path.

First using the formula of translational and radial acceleration the acceleration of the car is determined then in the next step the resultant acceleration of the car at point B is determined.

Fundamentals

The radial acceleration of an object is as:

Here is the velocity, is the radius of curvature and is the radial acceleration of the car.

The translational acceleration of the car in the derivative form in terms of velocity and distance is as:

Here is the translational velocity, is the translational acceleration, and is the distance.

The formula of the radius of the curvature of the radius is as:

Here is the displacement.

The differentiation of a variable with power with respect to is determined as:

And the differentiation of a constant with respect to is always zero as:

The formula to determine resultant acceleration for a curve path in which object has both translational and radial acceleration is as:

Here is the resultant acceleration.

At point B the acceleration of the car have the contribution of both translational acceleration and radial acceleration.

First determine the translational acceleration as:

Differentiate the velocity with respect to displacement using the concept of the differentiation as:

Substitute the velocity of the car after point A is

Substitute the velocity of car and the value of differentiation and the displacement to determine the translation acceleration as:

Now for the radial acceleration as:

The radius of the curvature for the given path is as:

As

Determine the first differential of as:

Substitute theas to determine:

Now differentiae to determine the as:

Substitute the and to determine the radius of the curvature as:

Since the velocity is:

Substitute the displacement as:

Therefore the radial acceleration will be as:

Substitute the value of velocity and the radius of curvature as:

The resultant acceleration of the car at point B from formula of resultant acceleration on curve path is:

Substitute the translation acceleration and radial acceleration as:

Ans:

The acceleration of the car at point B is.