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Video: Component Form of the Kinematics of an Accelerating Particle

Ed Burdette

The position of a particle is given by 𝑟(𝑡) = (4.0𝑡²𝑖 − 3.0𝑗 + 2.0𝑡³𝑘) m. What is the velocity of the particle at 0.0 s? What is the velocity of the particle at 1.0 s? What is the average velocity of the particle between 0.0 s and 1.0 s?

03:54

Video Transcript

The position of a particle is given by position as a function of time equals 4.0𝑡 squared in the 𝑖 direction minus 3.0 in the 𝑗 direction plus 2.0𝑡 cubed in the 𝑘 direction meters. What is the velocity of the particle at 0.0 seconds? What is the velocity of the particle at 1.0 seconds? What is the average velocity of the particle between 0.0 seconds and 1.0 seconds?

Let’s call the velocity of the particle at 0.0 seconds 𝑣 of 0.0 s and the velocity of the particle at 1.0 seconds 𝑣 of 1.0 s. And we’ll call the average velocity of the particle between 0.0 and 1.0 seconds 𝑣 sub avg.

Solving for these values will involve taking a time derivative of the position of the particle function. So let’s begin on that now.

We can recall that the velocity of a particle 𝑣 is equal to the time derivative of the position of the particle. Applying this relationship to our scenario, the velocity of the particle as a function of time is equal to the time derivative of 𝑟 of 𝑡.

We take that derivative term by term. The derivative of the 𝑖th component of 𝑟 is 8.0 times 𝑡; the derivative of the 𝑗th component of 𝑟 is zero; and the derivative of the 𝑘th component of 𝑟 is 6.0 times 𝑡 squared.

And because we’ve differentiated with respect to time, instead of units of meters we now have units of meters per second. We can now move ahead with solving for the velocity at 0.0 seconds.

When we plug in zero for 𝑡, all components of velocity go to zero, so the velocity at 0.0 seconds equals zero meters per second.

Now we can move onto solving for velocity when time equals 1.0 seconds. When we plug in 1.0 seconds for 𝑡, then our velocity equals 8.0𝑖 plus 6.0𝑘 meters per second.

Now we’re ready to move onto solving for 𝑣 sub avg. The particle’s average speed between 0.0 seconds and 1.0 seconds, that average speed is equal to the position of the particle at 1.0 seconds minus the position of the particle at 0.0 seconds divided by change in time which works out to 1.0 seconds.

If we plug in 1.0 seconds for 𝑡 in the 𝑟 of 𝑡 equation, then we find that’s equal to 4.0𝑖 minus 3.0𝑗 plus 2.0𝑘.

Then when we evaluate that function at time 𝑡 equals 0.0 seconds, it equals simply minus 3.0𝑗.

Since we’re subtracting that term in the first place, it becomes positive 3.0𝑗. Looking at the numerator of this fraction, we see that the 𝑗 component cancels out so that our final answer for the average speed of the particle over this time interval is 4.0𝑖 plus 2.0𝑘 meters per second.

This is the particle’s average velocity for the first second of its motion.