A 14.0 kg cart is moving with a velocity of 7.25 m/s down a level hallway. A constant force of -10.0 N acts on the cart, and its velocity becomes 3.20 m/s. Assume that air resistance is negligible.
(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?
final ke - initial ke = (1/2)14(3.2^2-7.25^2)
That is the work done
10 (distance) = work done
To solve this problem, we can use the equations for work, force, and change in kinetic energy.
(a) The change in kinetic energy of an object is determined by the equation ΔKE = KEf - KEi, where KEi is the initial kinetic energy and KEf is the final kinetic energy.
The initial kinetic energy can be calculated using the equation KEi = 1/2 * m * v^2, where m is the mass of the cart and v is the initial velocity.
KEi = 1/2 * 14.0 kg * (7.25 m/s)^2
KEi = 1/2 * 14.0 kg * 52.56 m^2/s^2
KEi = 366.36 J
The final kinetic energy can be calculated using the same equation, but with the final velocity.
KEf = 1/2 * 14.0 kg * (3.20 m/s)^2
KEf = 1/2 * 14.0 kg * 10.24 m^2/s^2
KEf = 71.68 J
Therefore, ΔKE = KEf - KEi = 71.68 J - 366.36 J = -294.68 J
The change in kinetic energy of the cart is -294.68 J.
(b) The work done on an object is determined by the equation W = F * d, where W is the work, F is the force applied, and d is the distance over which the force is applied.
In this case, the force applied is -10.0 N and we need to find the distance the cart moved while the force acted.
Let's assume the distance the cart moved is d.
Therefore, W = -10.0 N * d
Since the work done is equal to the change in kinetic energy (W = ΔKE), we can use the negative change in kinetic energy from part (a) to find the work done.
-294.68 J = -10.0 N * d
Solving for d:
d = (-294.68 J) / (-10.0 N)
d = 29.47 m
Therefore, the work done on the cart is -294.68 J, or 294.68 J of work is done on the cart.
(c) The distance the cart moved while the force acted is 29.47 meters.
To answer these questions, we need to use the equations of motion and the concept of work and energy.
(a) The change in kinetic energy can be calculated using the equation:
ΔKE = KE final - KE initial
Here, the initial kinetic energy (KE initial) is given by:
KE initial = (1/2) * mass * (velocity initial)^2
Substituting the given values:
mass = 14.0 kg
velocity initial = 7.25 m/s
KE initial = (1/2) * 14.0 kg * (7.25 m/s)^2
Next, we can calculate the final kinetic energy (KE final):
KE final = (1/2) * mass * (velocity final)^2
Substituting the given values:
velocity final = 3.20 m/s
KE final = (1/2) * 14.0 kg * (3.20 m/s)^2
Finally, we can calculate the change in kinetic energy:
ΔKE = KE final - KE initial
(b) The work done on an object is given by the equation:
Work = force * displacement * cos(θ)
Here, the force acting on the cart is -10.0 N (negative because it acts opposite to the motion).
The displacement (distance) is the distance traveled by the cart while the force is acting.
The angle between the force and the displacement is 0 degrees since they are in the same direction.
Therefore, the work done on the cart is:
Work = -10.0 N * displacement * cos(0) = -10.0 N * displacement * 1 = -10.0 N * displacement
(c) The distance traveled by the cart while the force is acting can be determined using the equation:
velocity final^2 = velocity initial^2 + 2 * acceleration * displacement
Here, the initial velocity (velocity initial) is 7.25 m/s.
The final velocity (velocity final) is 3.20 m/s.
The acceleration can be calculated using the equation:
acceleration = (force / mass)
Substituting the given values:
force = -10.0 N
mass = 14.0 kg
acceleration = (-10.0 N / 14.0 kg)
Now we can rearrange the equation to solve for displacement:
displacement = (velocity final^2 - velocity initial^2) / (2 * acceleration)
Substituting the given values, we can calculate the displacement.
These calculations will provide the answers to the questions.