Kinematics: notes and practice questions

(a) A delivery driver places a package on the flat trunk of their car. The car then accelerates forward.

State the nature and direction of the force that causes the package to accelerate with the car.

(b) The mass of the package is 25 kg, and the mass of the car is 1500 kg. The coefficient of static friction between the package and the car's trunk is 0.65.

Determine the maximum acceleration the car can have without the package sliding relative to the trunk.

(c) Calculate the maximum engine force that can be applied to the car for the package to remain stationary relative to the trunk. Assume no other horizontal forces act on the car.

An electric ski lift is powered by a motor at the base station. The motor is connected to a 750 V DC power supply by a cable with a total resistance of 0.15 Ω. When operating at full capacity, the motor draws a constant current of 400 A.

(a) Determine the potential difference across the terminals of the motor.

(b) The motor has an efficiency of 92%. Calculate the useful mechanical power output of the motor.

The ski lift carries skiers up a slope of length 1800 m that rises by a vertical height of 500 m. There are 50 chairs on the ascending side. Each empty chair has a mass of 25 kg and carries, on average, 1.5 skiers of average mass 75 kg. A constant resistive force of 12 kN opposes the motion.

(c) Determine the total upward force the motor must provide via the cable to maintain a constant speed.

(d) Estimate the maximum speed, $v$, of the ski lift.

(e) The lift operates continuously. Estimate the maximum number of skiers that can be transported to the top station in one hour.

(f) In an emergency stop, a brake is applied to a large solid steel disc, bringing the lift to a halt from its maximum speed. The total mass of the moving system (chairs, skiers, and cable) is 18 000 kg. Assume all the kinetic energy of the system is converted into thermal energy in the brake disc.

Calculate the temperature rise of the disc.

Data for this question:
Brake disc radius = 0.75 m
Brake disc thickness = 0.10 m
Density of steel = 7850 kg m⁻³
Specific heat capacity of steel = 450 J kg⁻¹ K⁻¹

(g) The speed of a chair is monitored using a radar device at the base station that emits microwaves of frequency 30 GHz. It measures the waves reflected from a chair as it moves away. The frequency shift, $Δf$, for a source moving directly away from a stationary observer can be approximated by the formula $Δf/f ≈ v/c$. In this radar measurement, this formula gives a good approximation for the shift detected.

Calculate the expected frequency shift.

A small drone is being tested. Its motion is tracked by a high-precision GPS system.

(a) The drone accelerates horizontally from rest. During this phase of motion, a student claims the drone has a changing speed but a constant velocity. Explain why this claim is incorrect.

(b) The drone is then programmed to fly in a perfect horizontal circle at a constant altitude and a constant speed of $5.0 \text{ m s}^{-1}$. Explain whether the drone has a constant velocity during this circular motion.

(c) In another test, the drone flies vertically upwards, slows to a stop at its maximum height, and then descends. Discuss the drone's velocity and acceleration at the instant it reaches its maximum height.

(d) A programmer suggests a flight path is possible where the drone moves with both a constant non-zero acceleration and a constant non-zero speed. Explain why this is not physically possible.

A spaceship departs from a space station, which is at rest in an inertial reference frame S. The spaceship travels at a constant velocity $v$. The clocks on the station and the spaceship are synchronized to zero at the moment of departure ($t = t' = 0$). At a later time, the spaceship emits a distress signal. According to an observer on the space station (frame S), this emission event occurs at a distance of 12 ly from the station and at a time of 13 years after departure.

(a) State what is meant by the spacetime interval between two events.

(b) Calculate the value of the spacetime interval between the departure of the spaceship and the emission of the signal, according to the observer on the space station.

(c) The emission of the signal occurs at the location of the spaceship. Determine the time elapsed on the spaceship's clock between departure and the emission of the signal.

(d) Explain why the time interval measured on the spaceship is known as the proper time interval.

(e) Calculate the velocity $v$ of the spaceship relative to the space station, as a fraction of the speed of light $c$.

The fine-structure constant, $\alpha$, is a fundamental physical constant that characterizes the strength of the electromagnetic interaction. It is given by the expression $\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}$.

(a) Show that the quantity $\frac{e^2}{4\pi\epsilon_0}$ has units of energy multiplied by distance.

(b) Hence, show that the fine-structure constant $\alpha$ is dimensionless.

A delivery drone makes a round trip from its base to a customer's location and back. The outbound journey to the customer is at a constant speed of $15.0 \text{ m s}^{-1}$. The return journey from the customer back to the base is at a constant speed of $25.0 \text{ m s}^{-1}$. The distance to the customer's location is the same as the distance back to the base.

(a) Calculate the average speed of the drone for the entire round trip.

(b) State the average velocity of the drone for the entire round trip.

(c) Explain why the average speed and average velocity for the entire round trip are different.

A drone is flying horizontally at a constant altitude of $80.0 \text{ m}$ above the ground with a velocity of $25.0 \text{ m s}^{-1}$. It releases a package intended for a target on the ground $100.0 \text{ m}$ ahead. Air resistance is negligible.

(a) Calculate the vertical distance the package drops by the time it reaches the horizontal position of the target.

(b) Calculate the total time taken for the package to hit the ground from the moment it is released.

(c) An identical package is held next to the drone and released from rest at the same instant the drone releases its package. Discuss which package, the one released from the drone or the one dropped from rest, will hit the ground first.

The drone is now used to launch a flare vertically upwards with an initial velocity of $60.0 \text{ m s}^{-1}$ from the same altitude of $80.0 \text{ m}$. Air resistance is negligible.

(d) (i) Calculate the maximum height the flare reaches above the launch point.

(d) (ii) Outline an assumption needed to answer (d)(i).

(d) (iii) Calculate the time taken for the flare to arrive back at the point of launching.

(a) A drone flies horizontally with a constant speed $v$ at a height $h$ above level ground. A package is released from the drone. Air resistance is negligible.

Show that the time $t$ taken for the package to reach the ground is given by $t = \sqrt{\frac{2h}{g}}$.

(b) The drone is flying at a speed of $18\text{ m s}^{-1}$ and the height $h$ is $32\text{ m}$.

Calculate the horizontal distance travelled by the package between its release and hitting the ground.

(c) Determine the vertical component of the velocity of the package just before it hits the ground.

A rescue helicopter is flying horizontally with a constant velocity $v$ at an altitude $H$ above level ground. A survival package is released from the helicopter. Air resistance is negligible.

(a) Deduce an expression, in terms of $H$ and the acceleration of free fall $g$, for the time $t$ it takes for the package to reach the ground.

(b) Determine an expression, in terms of $v$, $H$ and $g$, for the horizontal distance $R$ travelled by the package before it lands.

(c) Show that the magnitude of the vertical component of velocity, $v_y$, of the package just before it hits the ground is given by $v_y = \sqrt{2gH}$.

(d) The helicopter is flying at an altitude of $350 \text{ m}$ with a horizontal speed of $55 \text{ m s}^{-1}$. Calculate the speed of the package just before it strikes the ground. Use $g = 9.81 \text{ m s}^{-2}$.