Question Video: Finding the Velocity and Acceleration Vector of a Body given Its Position Vector in Three Dimensions

Find the velocity 𝑣(𝑡) and acceleration 𝑎(𝑡) of an object with the given position vector 𝑟(𝑡) = (3 cos 𝑡, 2 sin 𝑡, 1).

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Video Transcript

Find the velocity 𝑣 of 𝑡 and acceleration 𝑎 of 𝑡 of an object with the given position vector, 𝑟 of 𝑡 equals three cos 𝑡, two sin 𝑡, one.

In this exercise, we want to solve for 𝑣 as a function of 𝑡 and 𝑎 as a function of 𝑡. And to do that, we can recall relationships between position, velocity, and acceleration. If we start with position as a function of time, then the time derivative of that relationship is equal to velocity as a function of time. And if we take the time derivative of position a second time, then we’ll arrive at acceleration as a function of time. So starting with 𝑟 of 𝑡, we’ll take time derivatives of this expression in order to solve for 𝑣 of 𝑡 and 𝑎 of 𝑡.

First, we’ll calculate 𝑣 of 𝑡 by taking the time derivative of our expression. We recall that the derivative of cosine of 𝑡 is negative sine of 𝑡. And the derivative of sine of 𝑡 is positive cosine of 𝑡. And since the derivative of a constant is zero, we now have an expression for velocity as a function of time. This is 𝑣 of 𝑡.

Next, we move on to solving for acceleration. And acceleration is equal to the time derivative of velocity. That is equal to the time derivative of negative three sine 𝑡, two cosine 𝑡, zero. Once again, we use the trigonometric differentiation identities that the derivative of sine of 𝑡, with respect to 𝑡, is positive cosine of 𝑡. And the derivative of cosine of 𝑡 is negative sine of 𝑡. This differentiated velocity is our expression for acceleration.

Based on the position vector, we’ve now found velocity and acceleration.

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