PLEASE SHOW ALL WORK

A 15.0-kg cart is moving with a velocity of 7.10 m/s down a level hallway. A constant force of -5.0 N acts on the cart and its velocity becomes 3.00 m/s.

(a) What is the change in kinetic energy of the cart?

(b) How much work was done on the cart?

(c) How far did the cart move while the force acted?

Well, well, well, let's get to work with some physics calculations. Brace yourself for some fun!

(a) The change in kinetic energy is defined as the final kinetic energy minus the initial kinetic energy. We can calculate it using the formula:

ΔKE = 1/2 * m * (vf^2 - vi^2)

where m is the mass of the cart, vf is the final velocity, and vi is the initial velocity. Plugging in the values we have:

ΔKE = 1/2 * 15.0 kg * (3.00 m/s)^2 - 1/2 * 15.0 kg * (7.10 m/s)^2

ΔKE = 1/2 * 15.0 kg * 9 m^2/s^2 - 1/2 * 15.0 kg * 50.41 m^2/s^2

ΔKE = 1/2 * 135 kg m^2/s^2 - 1/2 * 756.15 kg m^2/s^2

ΔKE = (67.5 - 378.07) kg m^2/s^2

ΔKE = -310.57 kg m^2/s^2

So the change in kinetic energy of the cart is -310.57 kg m^2/s^2.

(b) The work done on an object is defined as the force multiplied by the distance over which the force is applied. In this case, the force is -5.0 N, and we need to find the distance traveled.

Since the force acting on the cart is constant, we can use the formula:

Work = force * distance

Plugging in the values we have:

Work = -5.0 N * distance

Now, to find the distance, we can use one of the kinematic equations. Let's use the equation:

vf^2 = vi^2 + 2ad

where vf is the final velocity, vi is the initial velocity, a is the acceleration (which we can find using Newton's second law), and d is the distance.

Solving for d, we have:

d = (vf^2 - vi^2) / (2a)

Let's find the acceleration first. Using Newton's second law:

force = mass * acceleration

-5.0 N = 15.0 kg * acceleration

acceleration = -5.0 N / 15.0 kg

acceleration = -0.3333 m/s^2

Now, let's calculate the distance:

d = (3.00 m/s)^2 - (7.10 m/s)^2 / (2 * -0.3333 m/s^2)

d = 9.00 m^2/s^2 - 50.41 m^2/s^2 / -0.6666 m/s^2

d = (-41.41 m^2/s^2) / -0.6666 m/s^2

d ≈ 62.13 m

So the distance traveled by the cart while the force acted is approximately 62.13 meters.

(c) Woohoo, we made it to the last question! The distance traveled by the cart while the force acted was approximately 62.13 meters. Cheers to that!

And there you have it, my friend! The change in kinetic energy of the cart is -310.57 kg m^2/s^2, the work done on the cart is -5.0 N * 62.13 m, and the cart traveled approximately 62.13 meters while the force acted. I hope this helps, and remember, physics can be quite the circus act sometimes!

To solve this problem, we can use the formulas for change in kinetic energy (ΔKE), work (W), and distance (d).

(a) The change in kinetic energy (ΔKE) can be calculated using the formula: ΔKE = KE_final - KE_initial.

Given:
Mass of the cart (m) = 15.0 kg
Initial velocity (v_initial) = 7.10 m/s
Final velocity (v_final) = 3.00 m/s

The initial kinetic energy (KE_initial) can be calculated using the formula: KE_initial = (1/2) * m * (v_initial)^2.

Substituting the given values, we have:
KE_initial = (1/2) * 15.0 kg * (7.10 m/s)^2

Calculate KE_initial.

(b) The work done (W) is given by the formula: W = F * d, where F is the force applied and d is the distance the cart moved.

Given:
Force (F) = -5.0 N

The work done (W) can be calculated using the above formula.

(c) The distance (d) can be calculated using the formula: d = v_initial * t, where t is the time the force acts.

To find the time (t), we can use the formula: t = (v_final - v_initial) / a, where a is the acceleration.

The acceleration (a) can be calculated using the formula: a = F / m, where F is the force applied and m is the mass of the cart.

Substituting the given values, we can find a.

Then, substitute a and the given values in the formula for t to find the time.

Finally, use the time (t) and the initial velocity (v_initial) in the formula for distance (d) to find the distance moved by the cart.

Now, let's solve each part step by step.

(a) Calculation of the change in kinetic energy (ΔKE):

KE_initial = (1/2) * 15.0 kg * (7.10 m/s)^2

Calculate KE_initial.

(b) Calculation of the work done (W):

W = F * d

Calculate W.

(c) Calculation of the distance (d):

Step 1: Calculate the acceleration (a):

a = F / m

Calculate a.

Step 2: Calculate the time (t):

t = (v_final - v_initial) / a

Calculate t.

Step 3: Calculate the distance (d):

d = v_initial * t

Calculate d.

To solve this problem, we will use the formulas for kinetic energy, work done, and distance traveled. Let's start with part (a):

(a) What is the change in kinetic energy of the cart?

The formula for kinetic energy is given by:

Kinetic energy = (1/2) * mass * velocity^2

Before the force was applied, the cart had a kinetic energy of:

Initial kinetic energy = (1/2) * mass * initial velocity^2

Given the mass (15.0 kg) and the initial velocity (7.10 m/s), we can calculate the initial kinetic energy:

Initial kinetic energy = (1/2) * 15.0 kg * (7.10 m/s)^2

Next, we need to calculate the final kinetic energy. The final kinetic energy is given by:

Final kinetic energy = (1/2) * mass * final velocity^2

Given the mass (15.0 kg) and the final velocity (3.00 m/s), we can calculate the final kinetic energy:

Final kinetic energy = (1/2) * 15.0 kg * (3.00 m/s)^2

Finally, the change in kinetic energy is given by the difference between the final and initial kinetic energies:

Change in kinetic energy = Final kinetic energy - Initial kinetic energy

Now, let's move on to part (b):

(b) How much work was done on the cart?

The work done on the cart can be calculated using the formula:

Work done = force * distance

Given the force (-5.0 N) and the change in velocity (3.00 m/s - 7.10 m/s), we can calculate the work done on the cart:

Work done = -5.0 N * (3.00 m/s - 7.10 m/s)

Finally, let's solve part (c):

(c) How far did the cart move while the force acted?

The distance traveled can be calculated using the formula:

Distance = (initial velocity + final velocity) / 2 * time

Given the initial velocity (7.10 m/s), final velocity (3.00 m/s), and the time, we can calculate the distance traveled:

Distance = (7.10 m/s + 3.00 m/s) / 2 * time

Unfortunately, the time is not provided in the problem statement, so we cannot find the exact distance traveled without that information.