Amechanic pushes a 2.50*10^3 kg car from rest to a speed of v, doing 5.00*10^3 Jof work in the process, During this time, the car moves 25.0m. Neglecting friction between car and road, find (a) v and (b) the horizontal force exerted on the car
b. W = F*d = 5,000,
F*25 = 5,000,
F = 200 N.
F = M*a = 200,
2500*a = 200,
a = 0.08 m/s^2.
a. V^2 = Vo^2 + 2a*d = 0 + 0.16*25 = 4,
V = 2 m/s.
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Now, let's solve the problem. We know that work done is equal to the change in kinetic energy:
Work = ΔKE
The kinetic energy of a moving object is given by:
KE = (1/2)mv^2
where m is the mass of the car and v is its final velocity.
(a) To find v, we'll first rearrange the equation for kinetic energy:
ΔKE = (1/2)mv^2
v^2 = (2ΔKE) / m
v = √[(2ΔKE) / m]
Plugging in the given values:
v = √[(2 × 5000 J) / 2500 kg]
v = √[2 J / kg]
v ≈ 1.414 m/s
So, the final velocity of the car is approximately 1.414 m/s.
(b) Now, let's calculate the horizontal force exerted on the car. We know that work done is equal to force times distance:
Work = Force × Distance
Therefore:
Force = Work / Distance
Force = 5000 J / 25.0 m
Force = 200 N
So, the horizontal force exerted on the car is 200 Newtons.
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To find the speed of the car (v) and the horizontal force exerted on the car, we can use the work-energy principle.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Mathematically, we can express this as:
Work = Change in Kinetic Energy
In this case, the work done on the car is given as 5.00*10^3 J. The initial kinetic energy of the car is zero as it starts from rest. The final kinetic energy is given by (1/2)mv^2, where m is the mass of the car and v is its speed.
Step 1: Calculate the final kinetic energy of the car
Given:
Mass of the car (m) = 2.50*10^3 kg
Work done on the car (W) = 5.00*10^3 J
By equating the work done on the car to the change in its kinetic energy, we have:
5.00*10^3 J = (1/2)(2.50*10^3 kg)v^2
Step 2: Solve for v
Rearranging the equation, we have:
v^2 = (2 * 5.00*10^3 J) / (2.50*10^3 kg)
v^2 = 10 J/kg
Taking the square root of both sides, we get:
v ≈ √10 m/s
Therefore, the speed of the car (v) is approximately 3.16 m/s.
Step 3: Calculate the horizontal force exerted on the car
To find the horizontal force exerted on the car, we can use Newton's second law of motion, which states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a). In this case, since the car moves horizontally at a constant speed, the net force (F) is zero.
However, there is an external force applied to the car in the form of the mechanic pushing it. This force (Fext) is equal to the work done on the car divided by the distance it moves.
Step 4: Calculate the external force exerted on the car
Given:
Distance moved by the car (d) = 25.0 m
Work done on the car (W) = 5.00*10^3 J
By dividing the work done on the car by the distance it moves, we have:
Fext = W / d
Fext = (5.00*10^3 J) / (25.0 m)
Fext = 200 N
Therefore, the horizontal force exerted on the car (Fext) is 200 N.
To find the answers, we need to use the work-energy theorem. According to the work-energy theorem, the work done on an object is equal to its change in kinetic energy.
(a) Let's find the final velocity (v) of the car.
Since the car is initially at rest, its initial kinetic energy is zero. The work done on the car is 5.00 * 10^3 J. So, we can set up the equation:
Work (W) = Change in Kinetic Energy (ΔKE)
5.00 * 10^3 J = (1/2) * m * v^2 - (1/2) * m * 0^2 (Using the formula for kinetic energy, KE = (1/2) * m * v^2)
Here, m is the mass of the car, which is 2.50 * 10^3 kg.
Simplifying the equation, we get:
5.00 * 10^3 J = (1/2) * (2.50 * 10^3 kg) * v^2
10 * 10^2 J = (2.50 * 10^3 kg) * v^2
40 J = v^2
To find v, take the square root of both sides:
v = √(40 J)
Calculating the square root of 40 J, we get the answer for (a) as the final velocity (v) of the car.
(b) To find the horizontal force exerted on the car, we can use the work-energy theorem again.
From the given information, we can calculate the work done by the horizontal force on the car. It is the same as the work done to move the car:
Work (W) = Force (F) * Distance (d)
5.00 * 10^3 J = F * 25.0 m
To find the horizontal force (F), rearrange the equation:
F = W/d
F = (5.00 * 10^3 J) / (25.0 m)
Calculating the division, we get the answer for (b) as the horizontal force exerted on the car.