A 25 kg box is being pushed across the floor by a constant force ‹ 100, 0, 0 › N. The coefficient of kinetic friction for the table and box is 0.15. At t = 6.0 s the box is at location ‹ 14, 3, −4 › m, traveling with velocity ‹ 6, 0, 0 › m/s. What is its position and velocity at t = 7.3 s?

Question

A 25 kg box is being pushed across the floor by a constant force ‹ 100, 0, 0 › N. The coefficient of kinetic friction for the table and box is 0.15. At t = 6.0 s the box is at location ‹ 14, 3, −4 › m, traveling with velocity ‹ 6, 0, 0 › m/s. What is its position and velocity at t = 7.3 s?

Answer 1

To find the position and velocity of the box at t = 7.3 s, we'll need to use the equations of motion. Let's break down the problem into steps:

Step 1: Calculate the acceleration of the box.
We can use Newton's second law to find the acceleration of the box. The net force acting on the box is the applied force minus the frictional force. The frictional force can be calculated using the coefficient of kinetic friction and the normal force.

Given:
Mass of the box (m) = 25 kg
Applied force (F) = ‹ 100, 0, 0 › N
Coefficient of kinetic friction (μk) = 0.15

Frictional force (Ffriction) = μk * Normal force

Normal force is equal to the weight of the box, which can be calculated as:
Weight (W) = mass (m) * acceleration due to gravity (g) ≈ 25 kg * 9.8 m/s^2

Now we can calculate the frictional force:
Ffriction = μk * W

Next, we can calculate the acceleration using Newton's second law:
Fnet = m * a

The net force is the applied force minus the frictional force:
Fnet = F - Ffriction

Finally, we can calculate the acceleration:
a = Fnet / m

Step 2: Calculate the displacement and change in velocity.
Given:
Initial position (x0) = ‹ 14, 3, -4 › m
Initial velocity (v0) = ‹ 6, 0, 0 › m/s
Time interval (Δt) = 7.3 s - 6.0 s = 1.3 s

Using the equations of motion:
Displacement (Δx) = v0 * Δt + (1/2) * a * Δt^2
Final position (x) = x0 + Δx

Change in velocity (Δv) = a * Δt
Final velocity (v) = v0 + Δv

Now, let's calculate each step.

Step 1: Calculate the acceleration.
Weight (W) = m * g ≈ 25 kg * 9.8 m/s^2
Normal force = W
Frictional force (Ffriction) = μk * Normal force
Net force (Fnet) = F - Ffriction
Acceleration (a) = Fnet / m

Step 2: Calculate the displacement and change in velocity.
Displacement (Δx) = v0 * Δt + (1/2) * a * Δt^2
Final position (x) = x0 + Δx
Change in velocity (Δv) = a * Δt
Final velocity (v) = v0 + Δv

Now, let's substitute the values into the equations and calculate the final position and velocity.