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Chapter 2: Problem 112

A minivan starts from rest on the road whose constant radius of curvature is \(40 \mathrm{m}\) and whose bank angle is \(10^{\circ} .\) The motion occurs in a horizontal plane. If the constant forward acceleration of the minivan is \(1.8 \mathrm{m} / \mathrm{s}^{2},\) determine the magnitude \(a\) of its total acceleration 5 seconds after starting.

Short Answer

Expert verified
The total acceleration of the minivan is approximately 2.71 m/s虏.

Step by step solution

01

Determine the Forward Velocity

Since the minivan starts from rest, its initial velocity is zero. Use the formula for linear acceleration, \( v = u + at \), where \( u = 0 \) m/s (initial velocity), \( a = 1.8 \) m/s\(^2\) (constant forward acceleration), and \( t = 5 \) s. Thus:\[v = 0 + 1.8 \, \times \, 5 = 9 \, \text{m/s}.\]
02

Calculate the Centripetal Acceleration

Centripetal acceleration is given by \( a_c = \frac{v^2}{r} \), where \( v = 9 \, \text{m/s} \) is the velocity calculated in Step 1, and \( r = 40 \, \text{m} \) is the radius of curvature. Substitute these values into the equation:\[a_c = \frac{9^2}{40} = 2.025 \, \text{m/s}^2.\]
03

Determine Total Acceleration

The total acceleration of the minivan is the vector sum of the forward acceleration \( a_t = 1.8 \, \text{m/s}^2 \) and the centripetal acceleration \( a_c = 2.025 \, \text{m/s}^2 \). Use the Pythagorean theorem to find the magnitude of the total acceleration:\[a = \sqrt{(1.8)^2 + (2.025)^2} \approx \sqrt{3.24 + 4.100625} \approx \sqrt{7.340625} \approx 2.71 \, \text{m/s}^2.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Acceleration
Centripetal acceleration occurs when an object moves along a curved path, such as a circle. It is always directed towards the center of the path's curvature. In our scenario with the minivan, centripetal acceleration plays a crucial role as the vehicle traverses a circular path with a radius of 40 meters.
The formula to calculate centripetal acceleration is \[ a_c = \frac{v^2}{r} \] where:
  • \( a_c \) is the centripetal acceleration,
  • \( v \) is the velocity of the object,
  • and \( r \) is the radius of the curvature.
In the example, when we calculated the minivan's velocity to be 9 m/s, we substituted this along with the radius \( r = 40 \text{ m} \) into the formula, yielding a centripetal acceleration of approximately 2.025 m/s虏. This acceleration is vital as it keeps the minivan on a curved path, otherwise, it would continue straight due to inertia.
Kinematic Equations
Kinematic equations are used to analyze the motion of objects in classical mechanics, especially when there is constant acceleration. These equations relate the quantities of initial velocity, final velocity, acceleration, time, and displacement. For the minivan:
We used the kinematic equation \[ v = u + at \] where:
  • \( v \) represents the final velocity,
  • \( u \) is the initial velocity,
  • \( a \) refers to the acceleration,
  • and \( t \) is the time over which the acceleration occurs.
Since the minivan starts from rest, its initial velocity \( u \) is zero. With a forward acceleration of 1.8 m/s虏 over 5 seconds, we derived the forward velocity as 9 m/s. Understanding this equation is key to solving many physics problems, as it allows us to determine how objects will move under a constant force.
Pythagorean Theorem
The Pythagorean theorem is a mathematical principle that relates the sides of a right triangle. It is crucial in physics when determining resultant vectors, such as total acceleration. When vectors are perpendicular, they can be thought of as forming a right triangle.
In this exercise, the total acceleration of the minivan is found by combining the straight-line (tangential) acceleration and the centripetal acceleration. We calculated this using:\[ a = \sqrt{(a_t)^2 + (a_c)^2} \]where:
  • \( a \) is the total acceleration,
  • \( a_t \) represents the tangential acceleration,
  • and \( a_c \) is the centripetal acceleration.
This approach is based on the property that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For the minivan, the calculated total acceleration was approximately 2.71 m/s虏, showing how both forms of acceleration contribute to its motion.

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Most popular questions from this chapter

Train \(A\) is traveling at a constant speed \(v_{A}=\) \(35 \mathrm{mi} / \mathrm{hr}\) while car \(B\) travels in a straight line along the road as shown at a constant speed \(v_{B}\). A conductor \(C\) in the train begins to walk to the rear of the train car at a constant speed of \(4 \mathrm{ft} / \mathrm{sec}\) relative to the train. If the conductor perceives car \(B\) to move directly westward at \(16 \mathrm{ft} / \mathrm{sec},\) how fast is the car traveling?

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On its takeoff roll, the airplane starts from rest and accelerates according to \(a=a_{0}-k v^{2},\) where \(a_{0}\) is the constant acceleration resulting from the engine thrust and \(-k v^{2}\) is the acceleration due to aerodynamic drag. If \(a_{0}=2 \mathrm{m} / \mathrm{s}^{2}, k=0.00004 \mathrm{m}^{-1}\) and \(v\) is in meters per second, determine the design length of runway required for the airplane to reach the takeoff speed of \(250 \mathrm{km} / \mathrm{h}\) if the drag term is \((a)\) excluded and (b) included

If the velocity \(v\) of a particle moving along a straight line decreases linearly with its displacement \(s\) from \(20 \mathrm{m} / \mathrm{s}\) to a value approaching zero at \(s=30 \mathrm{m}\) determine the acceleration \(a\) of the particle when \(s=15 \mathrm{m}\) and show that the particle never reaches the 30 -m displacement.
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