A robot used in a pharmacy picks up a medicine bottle at t = 0. It accelerates at 0.20 m/s² for 4.5 s, then travels without acceleration for 68 s and finally decelerates at ―0.40 m/s² for 2.5 s to reach the counter where the pharmacist will take the medicine from the robot. From how far away did the robot fetch the medicine?

검증된 단계별 안내

Step 1: Break the motion into three distinct phases: (1) acceleration, (2) constant velocity, and (3) deceleration. For each phase, calculate the distance traveled using the appropriate kinematic equations.

Step 2: For the acceleration phase, use the equation for distance under constant acceleration:

d = v i_{0} t + \frac{1}{2} a t^{2}

. Here,

v i_{0}

is 0 m/s,

a

is 0.20 m/s², and

t

is 4.5 s. Substitute these values to find the distance traveled during this phase.

Step 3: For the constant velocity phase, first calculate the velocity at the end of the acceleration phase using the equation

v = v i_{0} + a t

. Then, use the equation

d = v t

to calculate the distance traveled during the 68 s of constant velocity.

Step 4: For the deceleration phase, use the equation

d = v i_{0} t + \frac{1}{2} a t^{2}

. Here,

v i_{0}

is the velocity at the end of the constant velocity phase,

a

is -0.40 m/s², and

t

is 2.5 s. Substitute these values to find the distance traveled during this phase.

Step 5: Add the distances from all three phases to find the total distance traveled by the robot. This total distance represents how far away the robot fetched the medicine.

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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 problem, understanding how to calculate the distance traveled during different phases of motion (acceleration, constant speed, and deceleration) is essential for determining the total distance the robot traveled.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It can be positive (speeding up) or negative (decelerating). In this scenario, the robot experiences both positive acceleration (0.20 m/s²) and negative acceleration (−0.40 m/s²), which are crucial for calculating the distance covered during each phase of its journey.

Equations of Motion

The equations of motion relate an object's displacement, initial velocity, final velocity, acceleration, and time. These equations allow us to calculate the distance traveled during each phase of the robot's movement. For instance, the equation s = ut + 0.5at² can be used to find the distance during acceleration and deceleration, while the distance during constant speed can be calculated using s = vt.