A little red wagon with mass 7.0kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of4.00 m/s and then is pushed 3.0m in the direction of the initial velocity by a force with a magnitude of 10.0 N a.) Use the work-energy theorem to calculate the wagon's final speed b.) Calculate the acceleration produced by the force
To answer these questions, we will need to apply the work-energy theorem and Newton's second law of motion. Let's break down the steps:
a.) To calculate the wagon's final speed, we can use the work-energy theorem, which states that the work done on an object equals the change in its kinetic energy. In this case, the work done on the wagon is done by the force over the distance it is pushed.
The work done (W) is given by W = F * d * cos(theta), where F is the force applied, d is the distance, and cos(theta) is the angle between the force and the direction of motion. Since the force and motion are in the same direction, cos(theta) = 1.
Given:
Mass (m) = 7.0 kg
Initial speed (v1) = 4.00 m/s
Force (F) = 10.0 N
Distance (d) = 3.0 m
First, we need to find the work done by the force. The formula for work is W = F * d.
W = (10.0 N) * (3.0 m)
W = 30.0 J
The work done is equal to the change in kinetic energy. The initial kinetic energy (KE1) can be calculated as:
KE1 = (1/2) * m * v1^2
KE1 = (1/2) * 7.0 kg * (4.00 m/s)^2
KE1 = 56.0 J
Now, let's find the final kinetic energy (KE2) using the work-energy theorem equation:
KE2 = KE1 + W
KE2 = 56.0 J + 30.0 J
KE2 = 86.0 J
The final speed (v2) can be determined from the final kinetic energy:
KE2 = (1/2) * m * v2^2
86.0 J = (1/2) * 7.0 kg * v2^2
v2^2 = (86.0 J / (1/2 * 7.0 kg))
v2^2 = 24.5714 m^2/s^2
Taking the square root of both sides, we find:
v2 = √(24.5714 m^2/s^2)
v2 ≈ 4.96 m/s (approximately)
Therefore, the wagon's final speed is approximately 4.96 m/s.
b.) To calculate the acceleration produced by the force, we can use Newton's second law of motion, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration.
Given:
Mass (m) = 7.0 kg
Force (F) = 10.0 N
We can rearrange the formula to solve for acceleration (a):
F = m * a
a = F / m
a = 10.0 N / 7.0 kg
a ≈ 1.43 m/s^2 (approximately)
Therefore, the acceleration produced by the force is approximately 1.43 m/s^2.
a.) To calculate the wagon's final speed using the work-energy theorem, we need to calculate the work done on the wagon and equate it to the change in its kinetic energy.
The work done on the wagon is given by the formula: Work = Force * displacement * cos(theta), where theta is the angle between the force and the displacement.
Since the force and displacement are in the same direction, theta is 0 degrees and cos(theta) = 1.
Therefore, the work done on the wagon is: Work = Force * displacement.
Given that the force is 10.0 N and the displacement is 3.0 m, the work done on the wagon is: Work = 10.0 N * 3.0 m = 30.0 N·m or Joules (J).
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy.
So, the change in the wagon's kinetic energy is: Change in Kinetic Energy = Work = 30.0 J.
Since the initial speed of the wagon is 4.00 m/s and there is no work done against friction, the change in kinetic energy is also equal to the final kinetic energy minus the initial kinetic energy.
Therefore: Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy.
Substituting the values: 30.0 J = (1/2) * mass * (final speed)^2 - (1/2) * mass * (initial speed)^2.
Given that the mass of the wagon is 7.0 kg and the initial speed is 4.00 m/s, we can rearrange the equation to solve for the final speed.
(1/2) * 7.0 kg * (final speed)^2 = 30.0 J + (1/2) * 7.0 kg * (4.00 m/s)^2.
Simplifying the equation, we get:
(1/2) * 7.0 kg * (final speed)^2 = 30.0 J + 1/2 * 7.0 kg * 16.0 m^2/s^2.
(1/2) * 7.0 kg * (final speed)^2 = 30.0 J + 7.0 kg * 8.0 m^2/s^2.
(1/2) * 7.0 kg * (final speed)^2 = 30.0 J + 56.0 J.
(1/2) * 7.0 kg * (final speed)^2 = 86.0 J.
Multiplying both sides by 2 to isolate (final speed)^2, we get:
7.0 kg * (final speed)^2 = 172.0 J.
Dividing both sides by 7.0 kg, we get:
(final speed)^2 = 24.57 J/kg.
Taking the square root of both sides, we get the final speed:
final speed = √(24.57 J/kg) = 4.96 m/s.
Therefore, the wagon's final speed is 4.96 m/s.
b.) To calculate the acceleration produced by the force, we can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F = m * a).
The force applied to the wagon is given as 10.0 N, and the mass of the wagon is 7.0 kg.
Therefore: 10.0 N = 7.0 kg * a.
Dividing both sides of the equation by 7.0 kg, we get:
a = 10.0 N / 7.0 kg.
Calculating the value, we have:
a ≈ 1.43 m/s^2.
Therefore, the acceleration produced by the force is approximately 1.43 m/s^2.