Question

Students were discussing a problem in which the class was asked to find the acceleration of a cart rolling up and down an incline at the instant the cart was at the very top of its path. Malia says, At the top the velocity has to be zero so the acceleration has to be zero too. Sasha disagrees saying, No, the velocity is changing at the top so the acceleration cant be zero. Who do you agree with? Discuss the reason for your answer and explain why the one who is wrong might legitimately be confused about which answer is correct.

Answer 1

Sasha is correct: No, the velocity is changing at the top so the acceleration cant be zero.

Explanation:

The acceleration of an object is equal to the rate of change in velocity of the object:

$a=\frac{\Delta v}{\Delta t}$ (1)

where

$\Delta v$ is the change in velocity

$\Delta t$ is the time interval

For the cart on the ramp, as the cart reaches the top of the ramp, its velocity becomes temporarily zero:

v = 0

This is the origin of Malia's mistake.

However, this only lasts a moment; in reality, its velocity is changing, in particular its direction is changing (from upward to downward).

As we can see from eq.(1), a non-zero change in velocity implies a non-zero acceleration. Therefore, Sasha is right, since the cart has a non-zero acceleration.

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Answer 2

Sasha is right the velocity is changing at the top so the acceleration can't be zero.

The acceleration of an object is given by

a = Δv/Δt

where

Δv is the change in velocity

Δt is the time interval

Inclined surface:

• When the cart reaches the top of the inclined its velocity becomes zero in magnitude.
• This happens because a component of gravitational acceleration acts along the inclined plane against the motion of the cart going up.
• After the instant, the cart reaches the top and its velocity becomes zero, it begins to come down, which means that the direction of velocity changes at the top.
• The above evidence suggests that there is a non-zero acceleration at the top although the velocity is zero.

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