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Question 32 of 36

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SAT Subject Test

The figure shows a diagram of a box on a curved incline oriented from the upper left to the lower right.  The incline is steep at the top and becomes shallower toward the bottom where it smoothly transitions onto a horizontal surface. A solid-outline box labeled m with a 1 next to it is located near the top of the curved incline. A dashed-outline box with a 2 next to it is located near the bottom of the curved incline. To the left of the incline, a vertical measurement labeled h is shown that covers the vertical distance between box position one and box position 2. Along the curve, a measurement labeled d covers the curved distance measured along the incline between box position 1 and box position 2.

A box of mass m is released from rest at position 1 on the frictionless curved track shown above. It slides a distance d along the track in time t to reach position 2, dropping a vertical distance h. Let v and a be the instantaneous speed and instantaneous acceleration, respectively, of the box at position 2. Which of the following equations is valid for this situation?

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h equals v t

Correct Answer: 
No

h equals 1 half g t squared

Correct Answer: 
No

d equals 1 half a t squared

Correct Answer: 
No

v squared equals 2 a d

Correct Answer: 
No

m g h equals 1 half m v squared

Correct Answer: 
Yes

The correct answer is m g h equals fraction numerator 1 over denominator 2 end fraction m v to the power of 2 end exponent. This equation is an application of conservation of mechanical energy. In this situation the loss of potential energy mgh is equal to the gain of kinetic energy fraction numerator 1 over denominator 2 end fraction m v to the power of 2 end exponent. The other four equations are all kinematics equations that apply only when the acceleration is constant. Since the track is curved, the component of the gravitational force on the box that is tangent to the surface of the track decreases as the box slides. It is this component of the gravitational force that accelerates the box. By Newton’s second law, F equals m a, as the force decreases the acceleration also decreases. Since the acceleration is not constant, the other four equations are not valid.

Question Difficulty: 
hard