Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

문제 84b

2장, 문제 84b

A rubber ball is shot straight up from the ground with speed v₀. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. What is the maximum value of h for which a collision occurs before the first ball falls back to the ground?

검증된 단계별 안내

Define the motion equations for both balls. For the first ball (shot upwards), its position as a function of time is given by:

x_1(t) = v_0 t - \(\frac{1}{2}\) g t^2

, where

v_0

is the initial velocity and

g

is the acceleration due to gravity. For the second ball (dropped from height

h

), its position as a function of time is:

x_2(t) = h - \(\frac{1}{2}\) g t^2

Set the condition for collision. A collision occurs when the two balls are at the same position at the same time. This means

x_1(t) = x_2(t)

. Substitute the expressions for

x_1(t)

and

x_2(t)

into this equation:

v_0 t - \(\frac{1}{2}\) g t^2 = h - \(\frac{1}{2}\) g t^2

Simplify the equation. Cancel out the common term

-\(\frac{1}{2}\) g t^2

on both sides to get:

v_0 t = h

. Rearrange this equation to solve for

h

h = v_0 t

Determine the maximum time

t

for which the first ball is still moving upwards. The first ball reaches its highest point when its velocity becomes zero. Use the kinematic equation for velocity:

v = v_0 - g t

. Set

v = 0

to find the time to reach the highest point:

t = \(\frac{v_0}{g}\)

Substitute the maximum time

t = \(\frac{v_0}{g}\)

into the equation

h = v_0 t

to find the maximum height

h

for which a collision occurs:

h = v_0 \(\cdot\) \(\frac{v_0}{g}\) = \(\frac{v_0^2}{g}\)

. This is the maximum value of

h

for which the two balls collide before the first ball falls back to the ground.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

17m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, kinematic equations can be used to determine the positions of both balls over time, which is essential for analyzing their potential collision.

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, with no other forces acting on it. For the dropped ball, it accelerates downward at a constant rate of approximately 9.81 m/s². Understanding free fall is crucial for calculating how long it takes for the second ball to reach the ground and how high it can be dropped from to ensure a collision with the first ball.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the kinetic energy of the first ball when shot upwards will convert to potential energy at its peak height. This concept helps in determining the maximum height the first ball reaches, which is necessary to find the maximum height h for the second ball to ensure a collision.