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Zero acceleration can mean a nonzero falling speed

An idealized body falls vertically through still air. Take downward as positive, and let $v\ge0$ be its falling speed in metres per second. Assume an upward drag force of magnitude $cv^2$ , where $c=0.50\,\mathrm{N\,s^2/m^2}$ is held constant throughout the motion. The body has mass $m=2.0\,\mathrm{kg}$ . Use $g=10\,\mathrm{m/s^2}$ ; neglect buoyancy and all other forces. Find the terminal speed, meaning a constant falling speed at which acceleration is zero. Is the acceleration already zero when $v=4.0\,\mathrm{m/s}$ ? Predict the direction of acceleration before calculating.

Show a hint

Draw weight downward and drag upward. “Terminal” sets net force to zero; it does not set either the speed or the individual forces to zero.

Show the reasoning

1. Write the signed force equation. Let $a$ be acceleration, positive downward. Newton’s second law gives $ma=mg-cv^2$ . At $v=4.0\,\mathrm{m/s}$ , weight is $mg=20\,\mathrm N$ and drag is $0.50(4.0)^2=8.0\,\mathrm N$ . Thus $a=(20-8)/2=6.0\,\mathrm{m/s^2}$ downward: the body is still speeding up.

2. Set acceleration to zero for the terminal condition. If $v_T$ denotes terminal speed, $0=mg-cv_T^2$ , so

$v_T=\sqrt{\frac{mg}{c}}=\sqrt{40}\,\mathrm{m/s}\approx6.32\,\mathrm{m/s}.$

We take the positive root because $v_T$ is a downward speed. Its square has units $\mathrm N/(\mathrm{N\,s^2/m^2})=\mathrm{m^2/s^2}$ .

3. Check the balance and the interpretation. At that speed, drag is $0.50(40)=20\,\mathrm N$ , equal to weight. The forces remain present and the body continues falling. Above this speed the model gives upward acceleration, slowing a downward-moving body; below it the acceleration is downward. The assumed law describes this example, not every drag regime.

Your turn: change mass without changing drag geometry

Consider a body with the same shape, orientation and drag coefficient, but mass $m=4.5\,\mathrm{kg}$ . Keep the same assumptions and $g$ . Predict whether multiplying the mass by 2.25 also multiplies terminal speed by 2.25, then calculate the new terminal speed and compare it with the first one.

Check the scaling

The new weight is $4.5(10)=45\,\mathrm N$ . Balance gives $0.50v_T^2=45$ , hence $v_T=\sqrt{90}\,\mathrm{m/s}\approx9.49\,\mathrm{m/s}$ . Its ratio to the first speed is $\sqrt{90/40}=1.5$ .

With $c$ and $g$ fixed, terminal speed scales as the square root of mass: $\sqrt{2.25}=1.5$ . Check the force: $0.50(90)=45$ N. Scaling the speed by 2.25 would multiply drag by $2.25^2$ , which would not balance the new weight. Keeping the drag coefficient unchanged is essential; changing shape would require a new model.

Review force diagrams before using a drag law This prerequisite lesson helps with objects, axes and force components. It does not replace the drag model above.

A tutor can check your force directions and ask whether zero acceleration is an assumption or a result. Bring that decision with your calculation.

Related subject areas

Choose the area that matches the chapter you’re studying.

MechanicsElectricity & magnetismWaves, optics & modern physics

Common difficulties

Turning the situation into a model
Using vectors, signs, and units consistently
Selecting the relevant physical principle

How tutoring can help

We confirm support and availability for your chapter before booking. A tutor can help you distinguish acceleration at a stated speed from terminal speed, build the signed force equation and explain why balanced forces do not mean the body has stopped. You can work together from the drag example above or another permitted question.

Exam preparation for 203-SNV-MS

If drag is included in your assessment, solve the terminal-speed question with the solution closed. Substitute your speed into the assumed drag law and check that drag equals weight. Then retry with the changed mass, keeping the drag coefficient and gravitational acceleration fixed, and compare your result with your predicted speed ratio.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

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203-SNV-MS · official course reference