Step-by-Step Physics Problem Guidance: Kinematics, Dynamics, Circular Motion, Relative Velocity, and Projectile Motion

Q1. A particle starts at x = -2. From t = 0 to t = 3 s it accelerates at 2 m/s2. From t = 3 to t = 6 s it moves at a constant velocity. From t = 6 to t = 8 s it decelerates to a stop. What is its final position at t = 8 s?

Background

Topic: Kinematics (Motion with Constant Acceleration)

This question tests your understanding of motion in one dimension, including acceleration, constant velocity, and deceleration phases. You'll need to apply kinematic equations to piece together the particle's journey.

Key Terms and Formulas

• Displacement: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Final velocity: $v = v_0 + a t$

• Constant velocity motion: $x = x_0 + v t$

Step-by-Step Guidance

1. For the first interval (t = 0 to t = 3 s), identify the initial position ($x_0 = -2$), initial velocity ($v_0 = 0$), and acceleration ($a = 2$ m/s2). Use the displacement formula to find the position at t = 3 s.

2. Calculate the velocity at t = 3 s using $v = v_0 + a t$.

3. For the second interval (t = 3 to t = 6 s), the particle moves at the constant velocity found in the previous step. Use $x = x_0 + v t$ to find the position at t = 6 s.

4. For the third interval (t = 6 to t = 8 s), the particle decelerates to a stop. Use the kinematic equations to find the displacement during this interval, knowing the initial velocity (from t = 6 s) and that the final velocity is zero.

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Q2. A car passes a stationary police cruiser at a constant 30 m/s. The cruiser waits 1 s and then accelerates at a constant rate a. If the cruiser catches the car at the 300 m mark, what was the cruiser's acceleration?

Background

Topic: Kinematics (Relative Motion, Constant Acceleration)

This problem involves two objects: one moving at constant velocity and one starting from rest after a delay and accelerating. You'll need to set up equations for both and solve for the acceleration.

Key Terms and Formulas

• Displacement for constant velocity: $x = x_0 + v t$

• Displacement for constant acceleration: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

Step-by-Step Guidance

1. Write the equation for the car's position as a function of time, starting from $x_0 = 0$ and $v = 30$ m/s.

2. Write the equation for the cruiser's position, noting that it starts accelerating after 1 s (so its motion starts at $t = 1$ s).

3. Set the positions equal at the point where the cruiser catches up (at $x = 300$ m) and solve for the time it takes.

4. Substitute the time back into the cruiser's equation to solve for the acceleration $a$.

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Q3. Jupiter has a radius of 69,600 km and rotates once every 9.93 hrs. Calculate the tangential velocity and the centripetal acceleration of a person standing at a latitude of 65 degrees.

Background

Topic: Circular Motion

This question tests your ability to calculate tangential velocity and centripetal acceleration for a rotating sphere, considering the effect of latitude.

Key Terms and Formulas

• Tangential velocity: $v = \omega r'$

• Angular velocity: $\omega = \frac{2\pi}{T}$

• Effective radius at latitude $\lambda$: $r' = r \cos \lambda$

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

Step-by-Step Guidance

1. Convert Jupiter's radius to meters and the rotation period to seconds.

2. Calculate the angular velocity $\omega$ using $\omega = \frac{2\pi}{T}$.

3. Find the effective radius at 65 degrees latitude: $r' = r \cos(65^\circ)$.

4. Calculate the tangential velocity $v = \omega r'$.

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

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Q4. A fish swims in a river where the current flows due West at 16 mph. The fish's speed relative to the water is 22 mph. The fish wants to move in a direction such that its resultant path (relative to the ground) is pointed directly South. At what angle must the fish aim its body relative to the water to compensate for the current? What is the fish’s actual speed relative to the ground $V_{SG}$? The "Heading" Challenge: Translate your angle into a navigational heading (e.g., “theta degrees East of South").

Background

Topic: Relative Velocity (Vector Addition)

This problem involves vector addition to find the direction and magnitude of the fish's velocity relative to the ground, compensating for the river's current.

Key Terms and Formulas

• Relative velocity: $\vec{V}_{SG} = \vec{V}_{SW} + \vec{V}_{WG}$

• Use trigonometry to resolve vectors into components.

• Pythagorean theorem for resultant speed.

Step-by-Step Guidance

1. Draw a vector diagram showing the fish's velocity relative to the water and the water's velocity relative to the ground.

2. Set up the vector equation so that the resultant velocity points directly South (i.e., the West component cancels out).

3. Use trigonometry (sine and cosine) to find the angle the fish must aim relative to the water to cancel the Westward current.

4. Set up the calculation for the fish's actual speed relative to the ground using the Pythagorean theorem.

5. Translate the angle into a navigational heading (e.g., degrees East of South).

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Q5. A basketball is thrown from an initial height of 2 m toward a hoop 9.2 m away that is 3.1 m high. If the ball takes 1.6 s to reach the hoop, find the initial velocity magnitude $v_i$ and the launch angle $\theta$.

Background

Topic: Projectile Motion

This question tests your ability to analyze projectile motion in two dimensions, using kinematic equations to relate displacement, time, and initial velocity components.

Key Terms and Formulas

• Horizontal motion: $x = v_{i,x} t$

• Vertical motion: $y = y_0 + v_{i,y} t - \frac{1}{2} g t^2$

• Initial velocity components: $v_{i,x} = v_i \cos \theta$, $v_{i,y} = v_i \sin \theta$

Step-by-Step Guidance

1. Write the horizontal displacement equation: $x = v_i \cos \theta \cdot t$ and solve for $v_i \cos \theta$.

2. Write the vertical displacement equation: $y = y_0 + v_i \sin \theta \cdot t - \frac{1}{2} g t^2$ and substitute the known values.

3. Use the two equations to solve for $v_i$ and $\theta$ (set up the system but do not solve).

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