Question 1

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Projectile Motion — When Two Stones Reach the Same Height

Projectile motion often requires analyzing simultaneous vertical positions of two objects. In this case, we explored when two stones, thrown at different times and velocities, align at the same vertical height under gravity. This problem showcases how kinematics and algebra blend to solve real-world scenarios.

Equations of Motion

Stone A (launched at ##t=0##): ### y_A(t) = 18t - 4.9t^2 ###

Stone B (launched at ##t=2.5## s): ### y_B(t) = 32(t-2.5) - 4.9(t-2.5)^2 ###

Solving for Equal Heights

Accepted Answer

Set the equations equal: ### 18t - 4.9t^2 = 32(t-2.5) - 4.9(t-2.5)^2. ###

Question 2

Common Height

Accepted Answer

Substitute ##t=2.87## into Stone A’s equation: ### y_A(2.87) = 18(2.87) - 4.9(2.87)^2 \approx 11.4\;	ext{m}. ###

Question 3

Results

Meeting time: ##t \approx 2.87## s

Common height: ##y \approx 11.4## m

Key Takeaways

Model each projectile with its own kinematic equation.

Account for delayed launches using shifted time expressions.

Set vertical positions equal to find intersection times.

Verify with both equations to confirm consistency.

Summary Table

ParameterStone AStone BInitial Velocity (##v_0##)18 m/s32 m/sLaunch Time##t = 0## s##t = 2.5## sEquation of Motion##y_A(t) = 18t - 4.9t^2####y_B(t) = 32(t - 2.5) - 4.9(t - 2.5)^2##Meeting Time##t \approx 2.87## sCommon Height##y \approx 11.4## m

Similar Problems — Quick Meeting Time Calculations

These quick solutions illustrate how to determine the time when two vertically launched objects reach the same height under gravity. Each case considers different launch velocities and possible launch delays.

Problem 1

Accepted Answer

Two balls thrown upwards simultaneously with initial velocities ##v_{01} = 15## m/s and ##v_{02} = 20## m/s. Meeting time: ≈ 0.51 s

Question 4

Problem 2

Accepted Answer

Two rockets: Rocket A launched at ##t=0## with ##50## m/s, Rocket B launched 3 s later with ##60## m/s. Meeting time: ≈ 6.12 s

Question 5

Problem 3

Accepted Answer

Stone A launched at ##t=0## with ##20## m/s, Stone B launched at ##t=1## s with ##25## m/s. Meeting time: ≈ 2.72 s

Question 6

Problem 4

Accepted Answer

Two projectiles: velocities ##30## m/s and ##40## m/s, with a launch delay of 2 s. Meeting time: ≈ 4.39 s

Question 7

Problem 5

Accepted Answer

Two objects thrown with velocities ##25## m/s and ##35## m/s, second launched 1.5 s later. Meeting time: ≈ 3.63 s

JUPITER SCIENCE

Understanding Relative and Absolute Motion

Physics: Motion in One Dimension

Numerical Examples in Motion in One Dimension – Kinematics

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