Understanding Motion: Vectors, Scalars, and Acceleration

Consider a scenario where a projectile is launched at an angle $\theta$ with respect to the horizontal, experiencing air resistance proportional to the square of its velocity. Which of the following statements regarding its trajectory is most accurate?

A relativistic particle with rest mass $m_0$ is moving at a velocity $v = 0.8c$, where $c$ is the speed of light. What is the percentage increase in its observed mass compared to its rest mass?

A car accelerates uniformly from rest on a level surface. Neglecting air resistance and considering only the effects of static friction between the tires and the road, under what conditions will the maximum possible acceleration be achieved?

Consider two identical springs connected in series. If a mass $m$ is attached to the combination, the system oscillates with a frequency $f$. What would be the oscillation frequency if the same two springs were connected in parallel with the same mass $m$?

Two cars, A and B, are moving towards each other on a straight road. Car A has a velocity of $v_A$ and car B has a velocity of $v_B$. At time $t=0$, they are separated by a distance $d$. Both cars begin braking simultaneously, each with a constant deceleration of magnitude $a$. What is the minimum value of $d$ required to prevent a collision?

A block of mass $m$ slides down an inclined plane tilted at an angle $\theta$ with respect to the horizontal. The coefficient of kinetic friction between the block and the plane is $\mu_k$. What is the condition for the block to accelerate down the plane?

An object is launched vertically upwards with an initial velocity $v_0$ in a non-uniform gravitational field where $g(y) = g_0(1 + ky)$, with $y$ being the vertical distance from the launch point and $k$ a positive constant. What is the maximum height reached by the object?

A car is designed with crumple zones that allow for a maximum deceleration of $a_{max}$ during a collision. If the car, moving at an initial velocity $v_0$, undergoes a head-on collision, what is the minimum crumple distance $d$ required to ensure the deceleration does not exceed $a_{max}$?

Consider a classical system consisting of a particle of mass $m$ moving in a potential $V(x) = kx^4$, where $k$ is a positive constant. What is the approximate relationship between the average kinetic energy $T$ and the average potential energy $V$ for this system?

A car is equipped with an advanced anti-lock braking system (ABS) that modulates the braking force to maintain the wheels at the verge of slipping (i.e., maximizing static friction). How does this system primarily enhance car safety compared to a car without ABS during emergency braking scenarios?

A particle moves in a viscous medium, experiencing a drag force proportional to the square of its velocity ($F_d = -bv^2$). If the particle is released from rest under the influence of gravity ($F_g = mg$), determine the terminal velocity ($v_t$) of the particle.

A drag racer accelerates from rest with a constant acceleration $a$ over a quarter-mile (402.3 meters). Assuming negligible air resistance, what minimum average power must the engine deliver to achieve this, given the racer has a mass $m$?

A car's suspension system is modeled as a damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $b$. Determine the condition for the system to be critically damped, ensuring the quickest return to equilibrium without oscillation after a disturbance.

Imagine a modified Newton's cradle with four balls. Initially, balls 1 and 2 are swung together from the left, impacting balls 3 and 4 at rest. Assuming perfectly elastic collisions, what is the resulting motion of the balls after the impact?

A train is traveling at a constant velocity $v$ relative to the ground. A passenger walks from the back to the front of the train with a constant velocity $u$ relative to the train. Simultaneously, the train passes through a tunnel of length $L$. What is the time interval during which the passenger is inside the tunnel?

Consider a car moving on a circular track with radius $r$. The car starts from rest and its speed increases at a constant rate $a_t$ (tangential acceleration). Determine the angle $\theta$ (relative to the starting point) at which the magnitudes of the tangential and centripetal accelerations are equal.

A block of mass $m$ is attached to a spring with spring constant $k$ and oscillates on a horizontal frictionless surface. If the block is also subject to a velocity-dependent damping force $F = -bv$, what is the angular frequency ($\omega$) of the damped oscillations?

A car accelerates uniformly from rest to a speed $v$ in time $t$. What fraction of the total distance traveled during this acceleration is covered in the first half of the time interval ($t/2$)?

Two objects, A and B, of masses $m$ and $2m$ respectively, are connected by a massless string that passes over a frictionless pulley. Initially, object A is held at rest on a table, and object B hangs freely. If the system is released, what is the acceleration of object A across the table, assuming the table is frictionless?

A rocket in deep space (no external forces) expels exhaust gases at a constant velocity $v_e$ relative to the rocket. If the rocket's initial mass is $m_0$ and its final mass is $m_f$, what is the rocket's final velocity $v$ if it started from rest?

A car accelerates from rest with a constant tangential acceleration on a flat, horizontal, circular track. Which of the following statements accurately describes how the required coefficient of static friction between the tires and the road changes as the car's speed increases?

In a ballistic pendulum experiment, a bullet of mass $m$ is fired into a stationary block of mass $M$ suspended by a massless string. The bullet embeds itself in the block, and the block swings to a maximum height $h$. What is the initial speed of the bullet ($v$) before the collision?

A disc rolls without slipping down an inclined plane. What fraction of its total kinetic energy is rotational kinetic energy?

A car merges onto a highway, increasing its speed from $v_1$ to $v_2$ over a distance $d$ with non-constant acceleration. Assuming the acceleration varies linearly with distance, $a(x) = Ax + B$, determine the constants $A$ and $B$ in terms of $v_1$, $v_2$, and $d$.

A small block is placed on a rotating turntable at a distance $r$ from the center. The turntable's angular speed is slowly increased. If the coefficient of static friction between the block and the turntable is $\mu_s$, at what angular speed $\omega$ will the block begin to slip?

A uniform rod of length $L$ and mass $M$ is pivoted at one end. What is the period of small oscillations for this physical pendulum?

A projectile is launched with initial velocity $v_0$ at an angle $\theta$ above the horizontal from a height $h$ above the ground. If we consider air resistance to be negligible, what is the horizontal range $R$ of the projectile?

Consider a cyclist riding on a flat road. The cyclist exerts a constant force on the pedals. Which of the following is true regarding the cyclist's motion considering air resistance, which is proportional to the square of the velocity?

A car is traveling at a constant speed $v$ around a banked curve with a radius $r$. The banking angle is $\theta$. Assuming no friction, what is the correct expression for the speed $v$ that allows the car to navigate the curve without any reliance on friction?

A force $F(x) = -kx + bx^3$ acts on a particle, where $k$ and $b$ are positive constants. What is the nature of the equilibrium points for this system?

Two cars of equal mass approach an intersection. Car A is traveling east at 20 m/s and car B is traveling north at 30 m/s. The cars collide inelastically and stick together. What is the approximate speed and direction (relative to north) of the wreckage immediately after the collision?

A point mass $m$ is constrained to move without friction on the inside of a vertical, circular hoop of radius $r$. If the mass is given an initial velocity at the bottom of the hoop, what minimum speed is required for the mass to make a complete revolution without losing contact with the hoop?