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Chapter 2: Problem 20

When a ball is thrown up vertically with velocity \(v_{0}\), it reaches a maximum height of \(h\). If one wishes to triple the maximum height then the ball should be thrown with velocity (A) \(\sqrt{3} v_{0}\) (B) \(3 v_{0}\) (C) \(9 v_{0}\) (D) \(3 / 2 v_{0}\)

Short Answer

Expert verified
Using the equation of motion, we find the relationship between the initial velocity, maximum height, and gravitational acceleration: \(\displaystyle v_{0}^2 = 2gh\). To triple the maximum height, the new initial velocity \(v_1\) must satisfy \(\displaystyle v_1^2 = 3(v_{0}^2)\), which gives \(v_1 = \sqrt{3}v_{0}\). Therefore, the correct answer is (A) \(\displaystyle \sqrt{3} v_{0}\).

Step by step solution

01

We are given: 1. The initial velocity \(\displaystyle v_{0}\) 2. The maximum height \(\displaystyle h\) 3. The gravitational acceleration \(\displaystyle g\ ( approximately\ 9.81\ m/s^2)\) #Step 2#: Use the formula for the vertical motion

When the ball reaches its maximum height, its final velocity is 0. We can use the second equation of motion to relate initial velocity, maximum height, and gravitational acceleration: \[ v^2 = v_{0}^2 - 2gh \] At maximum height, \(\displaystyle v\ =\ 0\), so we have: \[ 0 = v_{0}^2 - 2gh \] #Step 3#: Solve for the initial velocity
02

Rearranging the above equation for initial velocity, we obtain: \[ v_{0}^2 = 2gh \] #Step 4#: Calculate the new velocity to triple the maximum height

Now, we want the new maximum height, \(\displaystyle h'\), to be triple the original maximum height, so \(\displaystyle h' = 3h\). Let the new initial velocity be \(\displaystyle v_1\), then we can write the equation for the new height: \[ v_1^2 = 2g(3h) \] #Step 5#: Relate the new initial velocity with the original initial velocity
03

Now, we want to express \(\displaystyle v_1\) in terms of \(\displaystyle v_{0}\). Using the equation \(\displaystyle v_{0}^2 = 2gh\), we can rewrite the equation for the new height as: \[ v_1^2 = 3( v_{0}^2) \] Then, taking the square root of both sides: \[ v_1 = \sqrt{3}v_{0} \] #Step 6#: Choose the correct answer

Comparing with the given choices, we can see that the correct answer is (A) \(\displaystyle \sqrt{3} v_{0}\).

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