Projectile Motion with Air Resistance
A student is investigating the motion of a small plastic ball launched horizontally from a tabletop. The ball leaves the table with a speed of 5.0 m/s. The table height is 1.2 m. In a vacuum, the ball would follow a parabolic path and land at a certain horizontal distance. However, in the laboratory, air resistance is not negligible. The student measures the actual horizontal range to be 1.8 m. The ball has a mass of 20 g and a cross-sectional area of 0.001 m². The density of air is 1.2 kg/m³. The drag coefficient is estimated to be 0.5. The student wants to understand the effect of air resistance on the motion and to compare the theoretical vacuum range with the measured range. Assume acceleration due to gravity is 9.8 m/s². The drag force is given by F_d = 0.5 * ρ * C_d * A * v², where ρ is air density, C_d is drag coefficient, A is cross-sectional area, and v is speed.
1. Calculate the horizontal range the ball would have in a vacuum. Show your steps and state any assumptions.
Read the worked answer
Step 1: Identify knowns.
Initial horizontal velocity, v_x = 5.0 m/s.
Initial vertical velocity, v_y0 = 0 m/s (launched horizontally).
Height, h = 1.2 m.
Acceleration due to gravity, g = 9.8 m/s².
Step 2: Calculate time of flight.
Vertical motion: h = 0.5 * g * t².
Rearrange: t = sqrt(2h/g) = sqrt(2 * 1.2 / 9.8) = sqrt(2.4/9.8) = sqrt(0.2449) ≈ 0.4949 s.
Step 3: Calculate horizontal range.
Horizontal motion: x = v_x * t = 5.0 * 0.4949 ≈ 2.4745 m.
Step 4: Round to appropriate significant figures.
Given data (5.0 m/s, 1.2 m) have two significant figures, so range ≈ 2.5 m.
Assumptions:
- No air resistance.
- The ball is a point particle (size negligible).
- The table is horizontal and the ball leaves exactly horizontally.
- g is constant at 9.8 m/s².
Answer: The vacuum range is approximately 2.5 m.
2. Estimate the initial drag force on the ball just after launch and compare it to the weight of the ball. Use this to explain why air resistance significantly affects the motion.
Read the worked answer
Step 1: Calculate initial speed.
Just after launch, speed v ≈ 5.0 m/s (horizontal).
Step 2: Calculate drag force.
F_d = 0.5 * ρ * C_d * A * v².
ρ = 1.2 kg/m³, C_d = 0.5, A = 0.001 m², v = 5.0 m/s.
F_d = 0.5 * 1.2 * 0.5 * 0.001 * (5.0)² = 0.5 * 1.2 * 0.5 * 0.001 * 25.
Compute: 0.51.2 = 0.6; 0.60.5 = 0.3; 0.30.001 = 0.0003; 0.000325 = 0.0075 N.
Step 3: Calculate weight.
Mass m = 20 g = 0.020 kg.
Weight W = m * g = 0.020 * 9.8 = 0.196 N.
Step 4: Compare.
Ratio F_d / W = 0.0075 / 0.196 ≈ 0.038, i.e., about 3.8% of the weight initially.
Although small, the drag force acts continuously and opposes motion, causing deceleration in both horizontal and vertical directions. Over the flight time (~0.5 s), the cumulative effect reduces the horizontal range from about 2.5 m to 1.8 m, a 28% decrease. This shows that even a small drag force can significantly alter the trajectory because it acts over the entire path and reduces both velocity components.
Answer: Initial drag force is about 0.0075 N, which is about 3.8% of the ball's weight (0.196 N). Despite being small, it causes a significant reduction in range due to its cumulative effect over time.
3. The measured range is 1.8 m. Using the vacuum range calculated earlier, determine the percentage reduction in range due to air resistance. Discuss whether the drag force remains constant during the flight and how that affects the accuracy of using a constant drag force to predict the range.
Read the worked answer
Step 1: Percentage reduction.
Vacuum range = 2.4745 m (using unrounded value).
Measured range = 1.8 m.
Reduction = 2.4745 – 1.8 = 0.6745 m.
Percentage reduction = (0.6745 / 2.4745) * 100 ≈ 27.3%.
Step 2: Variation of drag force.
Drag force depends on v². As the ball moves, its speed changes: initially 5.0 m/s horizontally, but as it falls, vertical velocity increases, so total speed increases (since v = sqrt(v_x² + v_y²)). However, drag opposes motion, so it reduces both components. The drag force is not constant; it decreases as the ball slows down horizontally but may increase initially as vertical speed builds up. The net effect is that drag is largest at the start when speed is highest, and decreases as the ball slows. Therefore, assuming a constant drag force based on initial speed would overestimate the average drag force, leading to an underestimated range. A more accurate prediction requires numerical integration or considering drag as velocity-dependent.
Step 3: Conclusion.
The range is reduced by about 27% due to air resistance. The drag force is not constant; it varies with speed, so a simple constant-force model is approximate. For precise work, one must account for the velocity dependence of drag.
Answer: The range is reduced by approximately 27.3%. The drag force is not constant; it varies with the square of speed, so using a constant drag force based on initial speed would not accurately predict the range. A more sophisticated model is needed for precise calculations.