Step-by-Step Guidance for Circular Motion and Relative Velocity Problems

Q1. An airplane flies between two points 500 km apart (destination directly north). The plane's airspeed is given, and a constant wind blows at 10.0 m/s due west. What direction must the plane fly relative to north to arrive at the destination?

Background

Topic: Relative Velocity in Two Dimensions

This question tests your understanding of how to combine velocity vectors (plane and wind) to determine the required heading for the plane to reach its destination.

Key Terms and Formulas

• Relative velocity: $\vec{v}_{\text{plane, ground}} = \vec{v}_{\text{plane, air}} + \vec{v}_{\text{wind}}$

• Vector components: Use trigonometry to resolve velocities into north/south and east/west components.

Step-by-Step Guidance

1. Draw a diagram showing the northward path, the westward wind, and the plane's velocity relative to the air.

2. Set up the vector equation: The plane's velocity relative to the ground must point directly north, so $\vec{v}_{\text{plane, ground}}$ has no east/west component.

3. Express the plane's airspeed as a vector at an angle $\theta$ east of north. Its east component must cancel the westward wind.

4. Write equations for the north and east components, and solve for $\theta$ using trigonometric relationships.

Try solving on your own before revealing the answer!

Q2. A plane has an eastward heading at a given speed (relative to air). A wind is blowing southward. What is the velocity of the plane relative to the ground?

Background

Topic: Vector Addition and Relative Velocity

This question tests your ability to add velocity vectors to find the resultant velocity and its direction.

Key Terms and Formulas

• Relative velocity: $\vec{v}_{\text{plane, ground}} = \vec{v}_{\text{plane, air}} + \vec{v}_{\text{wind}}$

• Magnitude and direction: $v = \sqrt{v_x^2 + v_y^2}$, $\theta = \arctan\left(\frac{v_y}{v_x}\right)$

Step-by-Step Guidance

1. Identify the eastward velocity of the plane and the southward velocity of the wind.

2. Write the velocity components: east ($v_x$) and south ($v_y$).

3. Add the components to get the resultant velocity vector.

4. Set up the formulas for magnitude and direction, but stop before calculating the final values.

Try solving on your own before revealing the answer!

Q3. A small boat moves at a given velocity when accelerated by a river current perpendicular to its initial direction. If the acceleration of the current is 0.750 m/s2, what will be the new velocity after a given time?

Background

Topic: Kinematics and Vector Addition

This question tests your ability to use kinematic equations and vector addition to find the resultant velocity after perpendicular acceleration.

Key Terms and Formulas

• Velocity after acceleration: $v = v_0 + a t$ (for each direction)

• Resultant velocity: $v_{\text{total}} = \sqrt{v_x^2 + v_y^2}$

• Direction: $\theta = \arctan\left(\frac{v_y}{v_x}\right)$

Step-by-Step Guidance

1. Identify the initial velocity and the direction of acceleration (perpendicular).

2. Calculate the change in velocity due to acceleration: $\Delta v = a t$.

3. Combine the initial velocity and the change in velocity as perpendicular components.

4. Set up the formulas for the magnitude and direction of the new velocity.

Try solving on your own before revealing the answer!

Q4. A ball is tied to the end of a cable and spun in a circle with a given radius, making 7.00 revolutions every given time. What is the magnitude of the acceleration of the ball?

Background

Topic: Uniform Circular Motion

This question tests your understanding of centripetal acceleration in circular motion.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Speed in circular motion: $v = 2 \pi r f$

• Frequency: $f = \frac{\text{revolutions}}{\text{time}}$

Step-by-Step Guidance

1. Calculate the frequency of revolution from the given data.

2. Find the speed of the ball using $v = 2 \pi r f$.

3. Set up the formula for centripetal acceleration $a_c = \frac{v^2}{r}$.

4. Plug in the values for $v$ and $r$, but stop before the final calculation.

Try solving on your own before revealing the answer!

Q5. A satellite orbits the earth at a distance of $1.50 \times 10^7$ m above the surface, taking a given time for each revolution. The earth's radius is given. What is the acceleration of the satellite?

Background

Topic: Circular Orbits and Centripetal Acceleration

This question tests your ability to calculate centripetal acceleration for an object in orbit.

Key Terms and Formulas

• Total orbital radius: $r = r_{\text{earth}} + h$

• Orbital speed: $v = \frac{2 \pi r}{T}$

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

Step-by-Step Guidance

1. Calculate the total orbital radius by adding the earth's radius and the altitude.

2. Find the orbital speed using the period and radius.

3. Set up the formula for centripetal acceleration.

4. Plug in the values, but stop before the final calculation.

Try solving on your own before revealing the answer!

Q6. A disk-shaped space station 175 m in diameter spins about its axis. How many rpm must it make so that the acceleration at the rim is $g/2$?

Background

Topic: Artificial Gravity and Rotational Motion

This question tests your ability to relate rotational speed to centripetal acceleration.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \omega^2 r$

• Angular speed: $\omega = 2 \pi f$

• Frequency: $f = \frac{\text{rpm}}{60}$

• Radius: $r = \frac{\text{diameter}}{2}$

Step-by-Step Guidance

1. Calculate the radius of the disk from its diameter.

2. Set up the equation for centripetal acceleration: $a_c = \omega^2 r$ and set $a_c = \frac{g}{2}$.

3. Express $\omega$ in terms of rpm and solve for rpm, but stop before the final calculation.

Try solving on your own before revealing the answer!