Question

Рух матеріальної точки заданий рівнянням x=4t+8r^2-16r^3. Визначити момент часу t, коли швидкість точки дорівнює нулю.Яким буде її прискорення а в цей момент?

Answer 1

Problem Analysis

We are given the equation of motion for a material point: x = 4t + 8r^2 - 16r^3. We need to determine the moment in time when the velocity of the point is zero, and also find the acceleration at that moment.

Finding the Moment When Velocity is Zero

To find the moment when the velocity of the point is zero, we need to find the derivative of the equation with respect to time (t) and set it equal to zero. This will give us the value of t at that moment.

Taking the derivative of x with respect to t: dx/dt = d(4t + 8r^2 - 16r^3)/dt = 4 + 0 - 0 = 4

Setting dx/dt equal to zero and solving for t: 4 = 0 t = 0

Therefore, the moment when the velocity of the point is zero is at t = 0.

Finding the Acceleration at that Moment

To find the acceleration at the moment when the velocity is zero, we need to find the second derivative of the equation with respect to time (t) and evaluate it at t = 0.

Taking the second derivative of x with respect to t: d^2x/dt^2 = d^2(4t + 8r^2 - 16r^3)/dt^2 = d(4)/dt = 0

Therefore, the acceleration at the moment when the velocity is zero is zero.

Conclusion

In summary, the moment when the velocity of the point is zero is at t = 0. At this moment, the acceleration of the point is zero as well.

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