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Chapter 10: Problem 54

Describe the motion of a particle if the tangential component of acceleration is \(0 .\)

Short Answer

Expert verified
The particle will move with a constant speed. If the path is a straight line, its velocity is also constant; on a circular path, it constantly changes direction due to radial (centripetal) acceleration.

Step by step solution

01

Description of Particle Motion

The particle moves with a constant speed, in a straight line or along the circumference of a circle. For linear motion, it means the velocity is constant and there isn't any acceleration. On a circular path, a non-zero radial acceleration keeps the particle moving in a circle, maintaining the constant speed. The direction of motion continually changes due to the radial acceleration.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Prove that the vector \(\mathbf{T}^{\prime}(t)\) is \(\mathbf{0}\) for an object moving in a straight line.

Use the model for projectile motion, assuming there is no air resistance. Rogers Centre in Toronto, Ontario has a center field fence that is 10 feet high and 400 feet from home plate. A ball is hit 3 feet above the ground and leaves the bat at a speed of 100 miles per hour. (a) The ball leaves the bat at an angle of \(\theta=\theta_{0}\) with the horizontal. Write the vector-valued function for the path of the ball. (b) Use a graphing utility to graph the vector-valued function for \(\theta_{0}=10^{\circ}, \theta_{0}=15^{\circ}, \theta_{0}=20^{\circ},\) and \(\theta_{0}=25^{\circ} .\) Use the graphs to approximate the minimum angle required for the hit to be a home run. (c) Determine analytically the minimum angle required for the hit to be a home run.

Use the model for projectile motion, assuming there is no air resistance. \([a(t)=-9.8\) meters per second per second \(]\) A projectile is fired from ground level at an angle of \(8^{\circ}\) with the horizontal. The projectile is to have a range of 50 meters. Find the minimum velocity necessary.

Find the indefinite integral. $$ \int\left[(2 t-1) \mathbf{i}+4 t^{3} \mathbf{j}+3 \sqrt{t} \mathbf{k}\right] d t $$

Find (a) \(\quad D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)] \quad\) and (b) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) by differentiating the product, then applying the properties of Theorem 10.2. $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad \mathbf{u}(t)=\mathbf{j}+t \mathbf{k} $$
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