Q: A particle moves in a vertical circle of radius $$R$$ under gravity. If its speed at the top of the circle is $$v$$, what is its speed at the bottom of the circle?

$$v_{\text{bot}} = \sqrt{v^2$ + 4gR}$$

Q: A boat crosses a river of width $$d$$ with speed $$v'$$ relative to the water, while the river current flows at speed $$v. If $$the crossing is perpendicular to the bank, what is the effective crossing speed?

$$v_{\text{net}} = \sqrt{(v')^2$ - v^2$}$$

Q: A boat crosses a river of width $$d at $$speed $$v'$$ perpendicular to the bank. What is the minimum crossing time?

$$t_{\text{min}} = \frac{d}{v'}$$

Question 1

A particle moves in a vertical circle of radius $$R$$ under gravity. If its speed at the top of the circle is $$v$$, what is its speed at the bottom of the circle?

Accepted Answer

$$v_{	ext{bot}} = \sqrt{v^2$ + 4gR}$$

Question 2

For a car moving in a vertical circle, what is the radial acceleration at the bottom of the circle if its speed at the top is $$v$$?

Accepted Answer

$$a_r(0) = \frac{v^2$}{R} + 4g$$

Question 3

A car maintains contact with a vertical circular track only if the normal force is non-negative. What is the minimum speed at the top of the circle required for this condition?

Accepted Answer

$$v_{	ext{min}} = \sqrt{gR}$$

Question 4

A boat crosses a river of width $$d$$ with speed $$v'$$ relative to the water, while the river current flows at speed $$v. If $$the crossing is perpendicular to the bank, what is the effective crossing speed?

Accepted Answer

$$v_{	ext{net}} = \sqrt{(v')^2$ - v^2$}$$

Question 5

Given two river crossings with times $$t_1$$ and $$t_2$$, and boat speeds $$v'_1$$ and $$v'_2 ($$relative to water), what equation allows you to solve for the river current speed $$v$$?

Accepted Answer

$$v = \sqrt{\frac{t_1$^2 (v'_1)^2$ - t_2$^2 (v'_2)^2$}{t_1$^2 - t_2$^2}}$$

Question 6

A rocket of mass $$m is $$descending from height $$h$$ with initial speed $$v$$ downward. It applies a constant upward thrust $$F to $$land softly (final speed zero). What is the required acceleration $$a$$ for this maneuver?

Accepted Answer

$$a = \frac{v^2$}{2h}$$

Question 7

For the rocket landing problem, what is the required thrust $$F to $$achieve a soft landing?

Accepted Answer

$$F = m \left( g + \frac{v^2$}{2h} \right)$$

Question 8

In the vertical circle problem, what is the expression for the normal force acceleration at an arbitrary angle $$\phi$$?

Accepted Answer

$$a_N(\phi) = \frac{v(\$$phi)^2$$}{R} + g \cos \phi$$

Question 9

A boat crosses a river of width $$d at $$speed $$v'$$ perpendicular to the bank. What is the minimum crossing time?

Accepted Answer

$$t_{	ext{min}} = \frac{d}{v'}$$

Question 10

During the minimum time river crossing, how far downstream does the boat drift due to the current?

Accepted Answer

$$x = v \cdot t_{	ext{min}}$$

Question 11

In the vertical circle scenario, what is the difference in normal force acceleration between the bottom and the top of the circle?

Accepted Answer

$$\Delta a_N = 6g$$

Question 12

A rocket of mass $$m is $$descending with speed $$v$$ from height $$h. If $$the rocket lands with zero speed, which kinematic equation can be used to relate acceleration $$a to$$ $$v$$ and $$h$$?

Accepted Answer

$$0^2$$ - $$(-v)^2 = 2a(-h$$)$$

Question 13

If a car moves in a vertical circle, what is the direction of the radial acceleration at the top and at the bottom?

Accepted Answer

At the top: downward (towards center); at the bottom: upward (towards center)

Question 14

In the river crossing problem, if the boat heads directly perpendicular to the bank, what is the effect of the river current on the crossing time?

Accepted Answer

The crossing time is unaffected by the current; only the drift downstream is affected.

Question 15

For a rocket landing problem, if the initial speed is doubled but the height remains the same, how does the required acceleration change?

Accepted Answer

The required acceleration increases by a factor of 4.

Question 16

In the vertical circle problem, what is the work-energy theorem used to relate the speed at different points?

Accepted Answer

$$\frac{1}{2} v(\$$phi)^2$$ - gR \cos \phi = 	ext{const}$$

Question 17

A boat crosses a river with a current. If the boat's speed relative to water is less than the current speed, what happens?

Accepted Answer

The boat cannot reach the opposite bank directly; it will be swept downstream.

Question 18

In the rocket landing scenario, which physical principle is used to relate the thrust force to the required acceleration?

Accepted Answer

Newton's Second Law: $$F = m(g + a)$$

Physics 1A: Circular Motion, Relative Motion, and Rocket Dynamics – Study Notes

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