📝 Free fall motion calculus (25 MCQs)

📖 From Calculus • 6. Integration • 25 questions available

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What is Free fall motion calculus?

Definition:
Free fall involves constant gravitational acceleration $g \approx 9.8 m/s^2$ downward. Equations are $a(t) = -g$ , $v(t) = -gt + v_0$ , $s(t) = -\frac{1}{2}gt^2 + v_0t + s_0$ , assuming upward is positive.

Example:
Drop ball from 100m. $v_0=0, s_0=100$ . $s(t) = -4.9t^2 + 100$ . Hits ground when $s=0$ : $4.9t^2=100 \Rightarrow t \approx 4.52 s$ .

Reason:
This model applies calculus to gravity, enabling accurate prediction of projectile motion, essential for engineering, sports science, and safety calculations.

Easy

Medium

Hard

📝 All Free fall motion calculus MCQs

Q1. A ball is thrown vertically upward with an initial velocity of 29.4 m/s. Approximately how long will it take for the ball to reach its maximum height? (Take g = 9.8 m/s²)

A.1.0 s

B.2.0 s

C.3.0 s ✅

D.6.0 s

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: At maximum height, the final velocity is 0 m/s. Using v = v₀ - gt, set 0 = 29.4 - 9.8t, which yields t = 29.4/9.8 = 3.0 s. This is a direct Easy of the constant acceleration formula. Distractor A might come from dividing by 29.4, B from a calculation error, and D from doubling the time.

Q2. A rock is dropped from rest and hits the ground with a speed of 24.5 m/s. What is the height from which it was dropped? (Take g = 9.8 m/s²)

A.15.0 m

B.30.6 m ✅

C.61.3 m

D.122.5 m

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using v² = v₀² + 2gh, with v₀ = 0, we get h = v²/(2g) = 24.5²/(2×9.8) = 600.25/19.6 = 30.625 m ≈ 30.6 m. Distractor A might come from h = v²/g, which lacks the factor ½. Distractor D might come from using v = gt and h = ½gt² incorrectly.

Q3. An object is launched vertically upward from ground level and returns to the ground. If the initial speed is doubled, the maximum height increases by a factor of:

A.2

B.4 ✅

C.8

D.√2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Maximum height h = v₀²/(2g) is proportional to the square of the initial velocity. Therefore, if initial speed doubles, h increases by a factor of 4. This is a conceptual reasoning question where the formula must be recognized. Distractor A comes from linear thinking, C from cubic thinking, and D from square root thinking.

Q4. A student claims that if you throw a ball upward with speed v, the time to go up is equal to the time to come back down. Is this always true?

A.Yes, because the acceleration is constant

B.Yes, because of the conservation of energy

C.No, if air resistance is ignored

D.No, if air resistance is present ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: In an ideal free-fall model (ignoring air resistance), the time up equals the time down due to symmetry. However, in the presence of air resistance, energy is lost, and the time down is longer. The student's claim is only valid in ideal conditions. The distractor C is incorrect because the model does ignore air resistance.

Q5. A ball is dropped from a height of 45 m. What is the speed of the ball just before it hits the ground? (Take g = 10 m/s²)

A.15 m/s

B.20 m/s

C.30 m/s ✅

D.40 m/s

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Using v² = 2gh = 2 × 10 × 45 = 900, so v = 30 m/s. This formula is derived from the constant acceleration equations. Distractor A comes from v = gt with t = 1.5s, B from a miscalculation, and D from v = gt with t = 4s.

Q6. The position function of a falling object is given by s(t) = 100 - 16t² (in feet). What is the velocity after 1 second?

A.32 ft/s

B.-32 ft/s ✅

C.16 ft/s

D.-16 ft/s

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Velocity is the derivative of position: v(t) = -32t. At t = 1, v(1) = -32 ft/s. The negative sign indicates downward motion. Distractor A ignores the negative sign, C might come from a misEasy of the derivative, and D from an incorrect derivative calculation.

Q7. A projectile is fired upward from the ground. At its highest point, which of the following is zero?

A.Velocity ✅

B.Acceleration

C.Both velocity and acceleration

D.Neither velocity nor acceleration

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At the maximum height, the velocity is momentarily zero. The acceleration is always g (downward), so it is not zero. This is a fundamental conceptual question. Distractor B confuses velocity and acceleration, C is incorrect because acceleration is never zero, and D ignores the fact that velocity is zero at the peak.

Q8. If an object is thrown upward with initial velocity v₀, the time to reach the maximum height is:

A.v₀/g ✅

B.2v₀/g

C.v₀/(2g)

D.g/v₀

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The time to reach maximum height is derived from v = v₀ - gt, setting v = 0. So t = v₀/g. This is a Easy of the formula. Distractor B is the total flight time, C is half the time, and D is the reciprocal of acceleration.

Q9. A coin is dropped from the top of a building. After 2 seconds, what is its displacement? (Take g = 10 m/s²)

A.10 m

B.20 m ✅

C.40 m

D.100 m

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using s = ½gt² = ½ × 10 × 4 = 20 m. This is a direct Easy of the displacement formula. Distractor A comes from s = gt, C from s = gt², and D from s = ½gt² with t = 10s.

Q10. Two balls are thrown upward with the same initial speed but from different heights. Which ball hits the ground with a greater speed?

A.The ball thrown from the higher height ✅

B.The ball thrown from the lower height

C.Both hit with the same speed

D.Cannot be determined without more information

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The impact speed is determined by v² = v₀² + 2gh. Since both have the same v₀, the one with the larger h (higher starting point) will have a greater final speed. Distractor C is common if one incorrectly assumes that initial speed is the only factor.

Q11. A ball is thrown upward and returns to the same height after 4 seconds. What was the initial velocity? (Take g = 10 m/s²)

A.10 m/s

B.20 m/s ✅

C.40 m/s

D.80 m/s

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Total flight time for a projectile returning to the same height is T = 2v₀/g. Therefore, v₀ = gT/2 = 10 × 4 / 2 = 20 m/s. Distractor A comes from T = v₀/g, C from v₀ = gT, and D from v₀ = 2gT.

Q12. A stone is thrown downward with an initial speed of 5 m/s from a height of 50 m. How long does it take to reach the ground? (Take g = 10 m/s²)

A.2.0 s

B.2.7 s ✅

C.3.0 s

D.3.7 s

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using s = v₀t + ½gt², we get 50 = 5t + 5t², which simplifies to t² + t - 10 = 0. Solving gives t ≈ 2.7 s. Distractor A comes from ignoring the initial velocity, C from a miscalculation, and D from using the wrong sign for v₀.

Q13. If an object is thrown upward with speed v, the maximum height reached is h. To double the maximum height, the initial speed must be:

A.Doubled

B.Quadrupled

C.Multiplied by √2 ✅

D.Multiplied by 4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Since h = v₀²/(2g), doubling h requires h′ = 2h = 2v₀²/(2g) = (√2 v₀)²/(2g). Thus, v₀ must be multiplied by √2. Distractor A comes from linear thinking, B from inverse relation, and D from cubic thinking.

Q14. Which of the following graphs best represents the position vs. time graph for a ball thrown upward and returning to the ground?

A.A straight line with positive slope

B.A parabola opening downward ✅

C.A parabola opening upward

D.A straight horizontal line

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The position vs. time graph for a projectile under constant acceleration is a parabola. Since the ball goes up and then down, the graph opens downward. Distractor A represents constant velocity, C represents acceleration upward, and D represents no motion.

Q15. A ball is dropped from rest. Which of the following correctly describes the sign of velocity and acceleration during the fall?

A.Both positive

B.Both negative

C.Velocity negative, acceleration negative ✅

D.Velocity negative, acceleration positive

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: If the positive direction is upward, the velocity is negative (downward) and the acceleration is also negative (downward due to gravity). Distractor A is incorrect because it assumes both are positive, B is incorrect because it assumes both are upward, and D is incorrect because it assumes the object is slowing down.

Q16. A student says that the acceleration of a ball thrown upward is zero at the highest point. Is this correct?

A.Yes, because the ball stops at the top

B.Yes, because there is no force at the top

C.No, acceleration is always g downward ✅

D.No, acceleration is zero only if there is no gravity

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The acceleration due to gravity is constant and always downward (g). It does not become zero at the highest point. The student's misconception arises from confusing velocity with acceleration. Distractor A is common, B is physically incorrect, and D is a misunderstanding of gravity.

Q17. A ball is thrown upward with an initial velocity of 15 m/s. If it is caught at the same height, what is its speed when caught? (Assume no air resistance)

A.0 m/s

B.15 m/s ✅

C.7.5 m/s

D.30 m/s

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: By conservation of energy (or symmetry of projectile motion), the speed when returning to the same height equals the initial speed. Distractor A is incorrect because speed is not zero at the top, C is half the speed, and D is double the speed.

Q18. A rock is thrown upward from a cliff. If the time to reach the maximum height is t₁ and the total time to hit the water below is t₂, which of the following is true?

A.t₁ < t₂ ✅

B.t₁ = t₂

C.t₁ > t₂

D.Cannot be determined

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Since the rock is thrown upward, it must travel up and then come back down past the starting point before hitting the water below. Thus, the total flight time t₂ must be greater than the time to reach the maximum height t₁. Distractor B would only be true if it returned to the cliff, C is impossible, and D ignores the physical reasoning.

Q19. If an object is thrown upward with initial velocity v₀, the maximum height h is given by:

A.v₀²/g

B.v₀²/(2g) ✅

C.2v₀²/g

D.v₀/(2g)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The maximum height is derived from v² = v₀² - 2gh, setting v = 0. So h = v₀²/(2g). This is a Easy of the formula. Distractor A is missing the factor ½, C has a factor 2, and D is the time to reach the top divided by g.

Q20. A ball is dropped from rest. In the first second, it falls h meters. In the next second, it falls:

A.h

B.2h

C.3h ✅

D.4h

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: In the first second: h = ½g(1)² = ½g. In the second second: distance = ½g(2² - 1²) = ½g(3) = 3h. This is a classic example showing that distances fallen in successive equal time intervals are in the ratio 1:3:5:... Distractor A ignores acceleration, B is linear thinking, and D is thinking it quadruples.

Q21. A ball is thrown upward with a speed of 20 m/s. At the maximum height, what is the acceleration?

A.0 m/s²

B.9.8 m/s² downward ✅

C.9.8 m/s² upward

D.Cannot be determined

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At the maximum height, the velocity is zero, but the acceleration is still 9.8 m/s² downward. Many students mistakenly think acceleration is zero at the top. Distractor A confuses velocity with acceleration, C has the wrong direction, and D is incorrect because the acceleration is always constant in free-fall.

Q22. Two objects are dropped from the same height, one with mass 1 kg and the other with mass 10 kg. Ignoring air resistance, which hits the ground first?

A.The 1 kg object

B.The 10 kg object

C.Both hit at the same time ✅

D.The heavier object hits first

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: In free-fall, all objects have the same acceleration g, regardless of mass. This is a fundamental principle of gravity. Distractor A and B assume mass affects motion, and D is a common misconception that heavier objects fall faster.

Q23. A projectile is fired upward. At which point is the speed zero?

A.At the starting point

B.At the maximum height ✅

C.Just before hitting the ground

D.At all points

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The speed of a projectile fired upward is zero at the maximum height because the vertical velocity becomes zero momentarily. Distractor A is incorrect because the initial speed is nonzero, C is incorrect because it hits the ground with speed, and D is impossible.

Q24. If a ball is thrown upward and returns to the ground, which of the following is true about the acceleration?

A.It is zero at the top

B.It is constant and downward ✅

C.It changes direction at the top

D.It is constant and upward

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: In free-fall, the acceleration due to gravity is constant and always directed downward. It does not change direction at the top. Distractor A is a common misconception, C is incorrect because acceleration does not change direction, and D has the wrong direction.

Q25. A ball is dropped from a height of 80 m. What is the time taken to hit the ground? (Take g = 10 m/s²)

A.2 s

B.4 s ✅

C.6 s

D.8 s

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using s = ½gt², we get 80 = ½ × 10 × t² → t² = 16 → t = 4 s. This is a direct Easy of the displacement formula. Distractor A comes from s = gt, C from a miscalculation, and D from using the wrong formula.

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