Math Problem Statement

A student on a bridge 200 π‘š above a dry river bed below drops a rock then

  1. 75 𝑠 later she throws a baseball straight down from the same height. The rock and baseball reach the ground at exactly the same time. a. What was the initial velocity of the baseball? b. What was the final velocity of the baseball (just as it reaches the ground but before any collision)? c. What was the final velocity of the rock (just as it reaches the ground but before any collision)?

Solution

To solve this problem, let's break it down step-by-step using the equations of motion under gravity. We have the following given information:

  • Height of the bridge, h=200h = 200 m
  • Acceleration due to gravity, g=9.8 m/s2g = 9.8 \, \text{m/s}^2
  • Time delay between dropping the rock and throwing the baseball, tdelay=1.75t_{\text{delay}} = 1.75 s

Part (a): Initial Velocity of the Baseball

  1. Time taken for the rock to hit the ground:

    • The rock is dropped from rest, so its initial velocity urock=0u_{\text{rock}} = 0.
    • Using the equation of motion:
      h=12gtrock2h = \frac{1}{2} g t_{\text{rock}}^2 Substituting the values: 200=12Γ—9.8Γ—trock2200 = \frac{1}{2} \times 9.8 \times t_{\text{rock}}^2 200=4.9Γ—trock2200 = 4.9 \times t_{\text{rock}}^2 trock2=2004.9t_{\text{rock}}^2 = \frac{200}{4.9} trock=2004.9t_{\text{rock}} = \sqrt{\frac{200}{4.9}}
  2. Time for the rock: Let's calculate this.

  3. Time taken for the baseball to reach the ground:

    • The baseball is thrown 1.75 seconds later, so its time to reach the ground: tbaseball=trockβˆ’1.75t_{\text{baseball}} = t_{\text{rock}} - 1.75
  4. Equation for the baseball's motion:

    • The baseball is thrown with an initial velocity ubaseballu_{\text{baseball}} straight down.
    • Using the equation: h=ubaseballtbaseball+12gtbaseball2h = u_{\text{baseball}} t_{\text{baseball}} + \frac{1}{2} g t_{\text{baseball}}^2 Substituting the known values, solve for ubaseballu_{\text{baseball}}.

Let's do these calculations step-by-step.1. Time taken for the rock to hit the ground:
trockβ‰ˆ6.39t_{\text{rock}} \approx 6.39 seconds

  1. Time taken for the baseball to reach the ground:
    tbaseballβ‰ˆ4.64t_{\text{baseball}} \approx 4.64 seconds

  2. Initial velocity of the baseball:
    ubaseballβ‰ˆ20.38 m/su_{\text{baseball}} \approx 20.38 \, \text{m/s}

Part (b): Final Velocity of the Baseball

To find the final velocity of the baseball just before it hits the ground, we use the equation of motion: vbaseball=ubaseball+gtbaseballv_{\text{baseball}} = u_{\text{baseball}} + g t_{\text{baseball}}

Let's calculate this value.The final velocity of the baseball just before it hits the ground is approximately 65.84 m/s65.84 \, \text{m/s}.

Part (c): Final Velocity of the Rock

For the rock, which was dropped (initial velocity urock=0u_{\text{rock}} = 0), the final velocity is given by: vrock=gtrockv_{\text{rock}} = g t_{\text{rock}}

Let's calculate this value.The final velocity of the rock just before it hits the ground is approximately 62.61 m/s62.61 \, \text{m/s}.

Summary

  • (a) Initial velocity of the baseball: 20.38 m/s20.38 \, \text{m/s}
  • (b) Final velocity of the baseball: 65.84 m/s65.84 \, \text{m/s}
  • (c) Final velocity of the rock: 62.61 m/s62.61 \, \text{m/s}

Do you want more details or have any questions about these calculations?

Related Questions

  1. What would be the initial velocity of the baseball if the rock was dropped from a different height?
  2. How would air resistance affect the velocities of both the baseball and the rock?
  3. What if the baseball were thrown upwards instead of downwards?
  4. How does the final velocity of the baseball change if thrown at an angle rather than straight down?
  5. How would the scenario change if the delay was longer or shorter than 1.75 seconds?

Tip

Always ensure to keep track of time and velocity directions when dealing with projectile motion to avoid sign errors in your equations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Kinematics
Projectile Motion
Gravity

Formulas

h = (1/2) * g * t^2
v = u + g * t
h = u * t + (1/2) * g * t^2

Theorems

Equations of motion under uniform acceleration

Suitable Grade Level

High School - Grades 11-12