A truck is traveling at a speed of 23.0 m/s along a level road. A crate is resting on the bed of the truck, and the coefficient of static friction between the crate and the truck bed is 0.610. Determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck.

force friction=mu(mg)

that equals ma so a=mu*g

vf^2=Vi^2+2ad put in a, find d

6.37

To determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck, we can use the equation:

\[d = \frac{v^2}{2 \mu g}\]

Where:
- \(d\) is the distance
- \(v\) is the initial velocity of the truck
- \(\mu\) is the coefficient of static friction
- \(g\) is the acceleration due to gravity (approximately 9.8 m/s^2)

Given:
- \(v = 23.0 \, \text{m/s}\)
- \(\mu = 0.610\)
- \(g = 9.8 \, \text{m/s}^2\)

Plugging in these values into the equation, we can calculate the shortest distance:

\[d = \frac{(23.0 \, \text{m/s})^2}{2 \times 0.610 \times 9.8 \, \text{m/s}^2}\]

First, let's calculate the numerator:

\((23.0 \, \text{m/s})^2 = 529.0 \, \text{m}^2/\text{s}^2\)

Now, let's calculate the denominator:

\(2 \times 0.610 \times 9.8 \, \text{m/s}^2 = 11.932 \, \text{m/s}^2\)

Now, let's divide the numerator by the denominator to find the shortest distance:

\[d = \frac{529.0 \, \text{m}^2/\text{s}^2}{11.932 \, \text{m/s}^2} = 44.29 \, \text{m}\]

Therefore, the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck is 44.29 meters.

To determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward, we need to determine the maximum static friction force that can be exerted on the crate.

First, we need to find the weight of the crate. Let's assume the mass of the crate is 'm'. The weight can be calculated using the formula: weight = mass * acceleration due to gravity (w = m * g), where the acceleration due to gravity is approximately 9.8 m/s^2.

Next, we calculate the maximum static friction force using the formula: f_max = coefficient of static friction * normal force (f_max = μ * N), where μ is the coefficient of static friction and N is the normal force. The normal force is equal to the weight of the crate when it's on a level road.

Since the crate is resting on the bed of the truck, the normal force is equal to the weight of the crate. Therefore, the normal force (N) is equal to the weight (w) calculated earlier.

Now, we can calculate the maximum static friction force (f_max) using the coefficient of static friction (μ) and the weight (w) of the crate.

Finally, to find the shortest distance in which the truck can come to a halt without causing the crate to slip forward, we use the formula: d = v^2 / (2 * a), where v is the initial velocity of the truck and a is the acceleration.

Substituting the values we have, we can calculate the shortest distance in which the truck can come to a halt without causing the crate to slip forward.