Math Problem Statement
The force exerted on a 8.0-kg block is shown in the figure as a function of time. How far does the block travel in the first 10.4 s? Assume that the motion is one dimensional and that the velocity of the block at time t= 0 s is 0 m/s.
Solution
To calculate the distance traveled by the 8.0-kg block in the first 10.4 seconds based on the force-time graph, we can follow these steps:
-
Find the acceleration as a function of time:
Use Newton's second law, , where is the force, is the mass, and is the acceleration. For this block: -
Find the velocity as a function of time:
The velocity is the integral of acceleration over time: Since the initial velocity is , the velocity at any time can be found by integrating the acceleration. -
Find the displacement:
The displacement is the integral of velocity over time:
Step-by-step process:
We will break down the time intervals based on the force changes in the graph:
- From 0 to 2 seconds: Force = 4 N
- From 2 to 4 seconds: Force = 8 N
- From 4 to 6 seconds: Force = 6 N
- From 6 to 9 seconds: Force = 4 N
- From 9 to 10.4 seconds: Force = 6 N
For each interval, we will calculate the acceleration, velocity, and displacement.
Step 1: Calculate accelerations for each interval.
- From s:
- From s:
- From s:
- From s:
- From s:
Let me now calculate the total displacement by integrating velocity over these intervals.The block travels a distance of approximately 37.39 meters in the first 10.4 seconds, and its velocity at the end of this time period is about 7.05 m/s.
Would you like further details on the calculations or the kinematic process?
Here are some related questions:
- How would the result change if the initial velocity was not zero?
- What would be the distance traveled if the force was constant?
- How can the work done on the block be calculated from the force-time graph?
- What is the average acceleration over the entire time period?
- How would friction affect the block's motion in this scenario?
Tip: Breaking complex problems into smaller intervals, like we did with time, often simplifies integration in physics problems.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Kinematics
Newton's Second Law
Integration
Piecewise Function
Formulas
F = ma
v(t) = v0 + ∫ a(t) dt
x(t) = ∫ v(t) dt
Theorems
Newton's Second Law of Motion
Suitable Grade Level
High School (Grades 9-12)
Related Recommendation
Acceleration and Velocity of a 1 kg Object with a Step Force Function
Find the Net Force at t = 1 s Using a Velocity-Time Graph
Calculate Displacement and Total Distance of a Block on a Frictionless Surface
Distance Calculation for a Car with Time-Dependent Acceleration
Calculate Average Force and Displacement from Velocity-Time Graph