Projectile Motion: Find Time Using Position Equation

What This Problem Teaches

• Writing kinematic equations from physical parameters
• Understanding the role of gravity in projectile motion
• Solving quadratic equations in physics contexts
• Interpreting multiple solutions and choosing the physically meaningful one
• Connecting algebraic expressions to real-world motion

Visualizing the Motion

Solution: Method 1 — The Kinematic Approach

Step 1 — Apply the kinematic position equation

For vertical motion under gravity, the position equation is:

where s₀ is initial height, v₀ is initial velocity, and a is acceleration due to gravity.

Step 2 — Substitute the known values

From the problem:

• Initial height: s₀ = 1 m
• Initial velocity: v₀ = 18 m/s (upward)
• Acceleration: a = -g = -9.8 m/s² (downward)

Step 3 — Set up the equation for the target height

To find when the bread reaches 16 meters, set s(t) = 16:

Step 4 — Solve using the quadratic formula

For equation at² + bt + c = 0 with a = -4.9, b = 18, c = -15:

Step 5 — Calculate both solutions

This gives us two times:

Step 6 — Choose the "way up" solution

Since we want the time on the way up, we need the earlier time: t ≈ 1.28 seconds

Solution: Method 2 — The Vertex Form Approach

Step 1 — Find the vertex of the parabola

The position equation s(t) = -4.9t² + 18t + 1 is a downward-opening parabola. Its vertex (maximum height) occurs at:

Step 2 — Determine which solution is "on the way up"

Since the maximum occurs at t ≈ 1.84 seconds, any time before this is "on the way up" and any time after is "on the way down."

Step 3 — Solve the quadratic as before

Setting -4.9t² + 18t + 1 = 16 gives us the same quadratic: -4.9t² + 18t - 15 = 0

Solutions: t ≈ 1.28 s and t ≈ 2.40 s

Step 4 — Select the upward solution

Since 1.28 < 1.84, the bread reaches 16 meters at t ≈ 1.28 seconds on the way up.

Verification

Let's check our answer by substituting t ≈ 1.278 back into the position equation:

The small difference from exactly 16 is due to rounding in our calculations. Our answer is correct.

Watch Out For These

The Pattern Behind This

All projectile motion problems follow the same kinematic template:

where g = -9.8 m/s² for Earth's surface gravity. The quadratic nature means objects pass through most heights twice: once going up, once coming down. The vertex of the parabola represents maximum height.

When solving for a specific height, you'll always get two solutions (unless you're solving for maximum height). The physical context tells you which one to choose - earlier time for "way up," later time for "way down."

Beyond the Textbook

Sports: Basketball players use projectile motion when shooting. The ball follows a parabolic path, and players instinctively adjust angle and velocity for different distances.

Engineering: Fountain designers use these equations to calculate water trajectory heights and distances, ensuring water lands in the intended basins.

Safety: Construction workers dropping tools (accidentally) can predict impact times and zones using projectile motion, crucial for jobsite safety protocols.

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