Quick answer: Terminal velocity is the steady speed at which a resistive force balances the weight, so the net force is zero. This AP Physics C worksheet on terminal velocity gives 20 problems from easy to hard: drag direction, units of coefficients and syrup terminal speed. The worked answers are on a separate page, and the worksheet is a free printable PDF.
These problems train the move from a velocity-dependent force to a differential equation and back to numbers. Early problems check units, directions and terminal speed; later ones ask for derivations with separation of variables, distances, and energy. They match AP Physics C: Mechanics unit 2 and first-year university mechanics.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 2
Problem 1 Drag direction ★★★
State Newton’s second law for an object that feels a resistive force, then say in which direction the drag force points compared with the velocity.1. Write the law as m dv/dt = ΣF.
2. Give the direction of drag.
Problem 2 Units of coefficients ★★★
A resistive force can be written F = bv or F = cv².1. Find the SI unit of b.
2. Find the SI unit of c.
Problem 3 Syrup terminal speed ★★★
Chiamaka in Lagos drops a marble of mass 0.20 kg into thick oil. The drag is F = bv with b = 0.50 kg/s.1. Calculate the terminal speed.
Problem 4 Time constant ★★★
For the marble in the previous problem (m = 0.20 kg, b = 0.50 kg/s):1. Calculate the time constant τ.
2. State what fraction of terminal speed the marble has reached after one time constant.
Problem 5 Paper cone ★★★
A paper cone of mass 4.0 g falls in air with F = cv², where c = 2.0 × 10⁻³ kg/m.1. Calculate its terminal speed.
Problem 6 Zero acceleration, nonzero speed ★★★
Mei in Singapore says: “At terminal velocity the object has stopped accelerating, so it must have stopped moving.”1. Explain what is wrong with this statement.
2. State what is true about the forces.
Problem 7 Speed in oil ★★★
A bead of mass 0.10 kg is released from rest in oil with b = 0.50 kg/s.1. Calculate vt and τ.
2. Calculate the speed 0.30 s after release.
Problem 8 Acceleration history ★★★
A sphere falls from rest through a fluid with time constant τ.1. Write the acceleration at t = 0.
2. Calculate the acceleration at t = τ.
3. Explain why the acceleration decreases.
Problem 9 Time to reach 90 % ★★★
A sphere has τ = 0.20 s in a thick liquid.1. Derive the time at which v = 0.90 vt.
2. Calculate its value.
Problem 10 Reading a graph ★★★
Aarav measures the speed of a 0.040 kg bead sinking in shampoo. Data: (t, v) = (0, 0), (0.25 s, 1.26 m/s), (0.50 s, 1.73 m/s), (0.75 s, 1.90 m/s), (1.0 s, 1.96 m/s), (1.5 s, 2.00 m/s). The points are plotted below.2. Estimate τ from the point where v is about 63 % of vt.
3. Calculate b.
Problem 11 Parachute constant ★★★
Diego in Texas descends under a parachute at a steady 6.0 m/s. His total mass is 75 kg and F = cv².1. Calculate c.
2. Calculate the drag force when his speed is 3.0 m/s.
Problem 12 Doubling the mass ★★★
Two identical hollow balls are joined, doubling the mass but keeping the same shape and cross-section.1. Predict the factor by which vt changes with linear drag.
2. Predict the factor with quadratic drag.
3. Explain the difference.
Problem 13 Spot the sign error ★★★
A student takes down as positive and writes m dv/dt = mg + bv for a falling ball.2. Write the correct equation.
3. State what the wrong equation would predict for the long-term speed.
Problem 14 Verify the solution ★★★
Show that v = vt(1 − e−t/τ), with vt = mg/b and τ = m/b, satisfies m dv/dt = mg − bv.1. Differentiate v.
2. Substitute into both sides.
Problem 15 Distance fallen ★★★
Wanjiru in Nairobi studies a bead with vt = 1.5 m/s and τ = 0.30 s starting from rest.1. Integrate v(t) to obtain x(t).
2. Calculate the distance fallen in the first 0.30 s.
Problem 16 Boat coasting to rest ★★★
Bilal in Lahore switches off a 400 kg boat’s engine at 6.0 m/s. Water drag is F = bv with b = 50 kg/s. Ignore all other forces.1. Show that v = v₀e−bt/m.
2. Calculate the speed after 12 s.
3. Calculate the total distance travelled before it stops.
Problem 17 Skydiver with tanh ★★★
A skydiver has vt = 30 m/s and v = vt tanh(gt/vt).1. Calculate the speed after 3.0 s.
2. Calculate the speed after 3.0 s if there were no drag.
3. Comment on the difference.
Problem 18 Ball thrown upward ★★★
A ball is launched upward at 25 m/s. Drag is F = cv² with c/m = 2.0 × 10⁻³ m⁻¹.2. Calculate the terminal speed on the way down.
3. Using v dv/dx = −g − (c/m)v², show that the maximum height is H = (m/2c) ln(1 + cv₀²/mg) and calculate it.
Problem 19 Block on an oiled ramp ★★★
A 2.0 kg block slides down a ramp inclined at 30° on a thin oil film that gives drag F = bv with b = 5.0 kg/s. Ignore solid friction.2. Calculate the terminal speed.
3. Calculate τ.
Problem 20 Power at terminal speed ★★★
Kofi’s skydiver (m = 80 kg, vt = 62.6 m/s) falls steadily.1. Explain why the power dissipated by drag equals the rate of loss of gravitational potential energy.
2. Calculate the power dissipated by drag.
