Question

The motorcycle is traveling at a constant speed of 76km/h. Determine the magnitude of its acceleration when it is at point A.

The motorcycle is traveling at a constant speed of 76km/h.
The motorcycle is traveling at a constant speed of 76km/h.

Answer

The magnitude of its acceleration when it is at point A is 1.223 m/s2.

Explanation

Data: The information from the question

Constant speed, v = 76 km/h = (76 x 1000) / (60 x 60) = 76000 / 3600 = 21.1 m/s

The tangential acceleration, at = dv/dt = 0

x = 25 m

We also have y2 = 2x. Thus, we can take the square root of both sides to end up with y = √2x

But since x = 25, we can then say that y = √2x = √2 x 25 = √50. Therefore, y = √50.

Unknown: The information we need to find

The normal acceleration, an = ?

Formula: The equation that will help us to solve the problem

First step: We will apply the first and second derivatives of 2x

Second Step: We will use the product rule

Third step: We will now employ the equation for the radius of the curvature ρ = [1 + (dy/dx)2]3/2 / (d2y/dx2)

Fourth Step: We will finally apply the formula an = v2 / ρ

Solution

First step:

y2 = 2x

After differentiating the above equation, we will now have

2y (dy/dx) = 2

We can now divide both sides by 2y to obtain

(dy/dx) = 1/y

Second step:

We will now use the product rule

2y (d2y/dx2) + 2 (dy/dx)2 = 0

After collecting like terms and dividing both sides by 2y, we will get:

(d2y/dx2) = – (1/y)(dy/dx)2

We can now substitute our data into the above equation

(d2y/dx2) = – (1/√50)(dy/dx)2

But dy/dx = 1/√50

Therefore,

(d2y/dx2) = – (1/√50)(1/√50)2

We will now have

(d2y/dx2) = (1/(50√50))

(d2y/dx2) = (1/353.6) = 0.00283 = 2.83 x 10-3

Remember that dy/dx 1/√50 = 0.141

Third step:

We will now use the formula ρ = [1 + (dy/dx)2]3/2 / (d2y/dx2) to find the radius of the curvature (ρ).

ρ = [1 + (dy/dx)2]3/2 / (d2y/dx2) = [1 + (0.141)2]3/2 / (2.83 x 10-3) = 363.95 m

Fourth step:

The final step is to use the formula an = v / ρ to find the magnitude of the acceleration. Therefore

an = v2 / ρ = (21.1 m/s)2 / (363.95 m) = 445.21 m2/s2 / 363.95 m = 1.223 m/s2

Therefore, the magnitude of the acceleration is 1.223 meters per second square (m/s2)

You may also like to read:

How to Find Acceleration

Reference:

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