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Chapter 2: Problem 17

A student standing on the ground throws a ball straight up. The ball leaves the student's hand with a speed of \(15 \mathrm{m} / \mathrm{s}\) when the hand is \(2.0 \mathrm{m}\) above the ground. How long is the ball in the air before it hits the ground? (The student moves her hand out of the way.)

Short Answer

Expert verified
The ball is approximately 2.61 seconds in the air before it hits the ground.

Step by step solution

01

Identify Relevant Physical Principles and Variables

For a ball thrown vertically upward, the physics principles involved are kinematics, specifically the equations of motion in vertical direction. Gravity is the only force acting on the ball and thus it accelerates downwards. We have the initial velocity (Vo) of 15 m/s, acceleration due to gravity (g) -9.8 m/s虏 (negative because it is acting downwards), the initial height (h) of 2.0 m above the ground, and the final position (y) of the ball is 0 (at ground level).
02

Equation of Motion

We use the kinematic equation: \(y = V_ot + 0.5gt^2\), where V= initial velocity; g = acceleration due to gravity, t = time, and y = final position.
03

Insert Known Values into the Equation

We replace in the equation y, Vo, g giving us: 0 = 15t + 0.5*(-9.8)*t虏 + 2. This becomes a quadratic equation. Which can be written as \(4.9t^2 - 15t -2 = 0\)
04

Solving for Time

We use the quadratic formula \(t = [-b 卤 sqrt(b虏-4ac)] / 2a\), where a = 4.9, b = -15, and c = -2. Then, calculate the two possible values of t. Remember, time cannot be negative, so the physical solution corresponds to the positive time value.

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Most popular questions from this chapter

Julie drives 100 mi to Grandmother's house. On the way to Grandmother's, Julie drives half the distance at 40 mph and half the distance at 60 mph. On her return trip, she drives half the time at 40 mph and half the time at 60 mph. a. What is Julie's average speed on the way to Grandmother's house? b. What is her average speed on the return trip?

You are given the kinematic equation or equations that are used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation(s). Be sure that the answer your problem requests is consistent with the equation(s) given. b. Draw the pictorial representation for your problem. c. Finish the solution of the problem. $$64 \mathrm{m}=0 \mathrm{m}+(32 \mathrm{m} / \mathrm{s})(4 \mathrm{s}-0 \mathrm{s})+\frac{1}{2} a_{x}(4 \mathrm{s}-0 \mathrm{s})^{2}$$

David is driving a steady \(30 \mathrm{m} / \mathrm{s}\) when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady \(2.0 \mathrm{m} / \mathrm{s}^{2}\) at the instant when David passes. a. How far does Tina drive before passing David? b. What is her speed as she passes him?

A car starts at the origin and moves with velocity \(\vec{v}=\) \((10 \mathrm{m} / \mathrm{s},\) northeast). How far from the origin will the car be after traveling for 45 s?

Careful measurements have been made of Olympic sprinters in the 100 -meter dash. A quite realistic model is that the sprinter's velocity is given by $$v_{x}=a\left(1-e^{-b t}\right)$$ where \(t\) is in \(\mathrm{s}, v_{x}\) is in \(\mathrm{m} / \mathrm{s},\) and the constants \(a\) and \(b\) are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with \(a=11.81 \mathrm{m} / \mathrm{s}\) and \(b=0.6887 s^{-1}\) a. What was Lewis's acceleration at \(t=0 \mathrm{s}, 2.00 \mathrm{s},\) and \(4.00 \mathrm{s} ?\) b. Find an expression for the distance traveled at time \(t\) c. Your expression from part \(\mathbf{b}\) is a transcendental equation, meaning that you can't solve it for \(t .\) However, it's not hard to use rial and error to find the time needed to travel a specific distance. To the nearest \(0.01 \mathrm{s}\), find the time Lewis needed to sprint \(100.0 \mathrm{m} .\) His official time was \(0.01 \mathrm{s}\) more than your answer, showing that this model is very good, but not perfect.
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