Force: Mathematical problem & solution-1

Problem:

A car with a mass of $1500$ kg accelerates uniformly from rest to a speed of $25$ m/s over a distance of $200$ meters. Calculate:

1. The acceleration of the car.
2. The net force acting on the car.
3. The work done by the net force.
4. The power developed by the net force at the final speed.

Solution:

1. Calculate the Acceleration

We can use the kinematic equation that relates initial velocity ($𝑢$), final velocity ($𝑣$), acceleration ($𝑎$), and distance ($𝑠$):

$${𝑣}^2={𝑢}^2+2𝑎𝑠$$

Given:

• Initial velocity, $𝑢=0$ (starting from rest)
• Final velocity, $𝑣=25$ m/s
• Distance, $𝑠=200$ m

Rearranging the equation to solve for $𝑎$:

$${𝑣}^2=2𝑎𝑠⟹𝑎=\frac{{𝑣}^2}{2𝑠}$$

Substituting the given values:

$$𝑎=\frac{(25\text{ m/s}{)}^2}{2\times 200\text{ m}}=\frac{625{\text{ m}}^2\mathrm{/}{\text{s}}^2}{400\text{ m}}=1.5625{\text{ m/s}}^2$$

So, the acceleration of the car is $1.5625{\text{ m/s}}^2$.

2. Calculate the Net Force

Using Newton's second law, the net force ($𝐹$) acting on the car is:

$$𝐹=𝑚𝑎$$

Given:

• Mass, $𝑚=1500$ kg
• Acceleration, $𝑎=1.5625{\text{ m/s}}^2$

$$𝐹=1500\text{ kg}\times 1.5625{\text{ m/s}}^2=2343.75\text{ N}$$

So, the net force acting on the car is $2343.75\text{ N}$.

3. Calculate the Work Done by the Net Force

The work done ($𝑊$) by the net force is the product of the force and the distance over which the force is applied:

$$𝑊=𝐹\cdot 𝑠$$

Given:

• Force, $𝐹=2343.75$ N
• Distance, $𝑠=200$ m

$$𝑊=2343.75\text{ N}\times 200\text{ m}=468750\text{ J}$$

So, the work done by the net force is $468750\text{ J}$ (Joules).

4. Calculate the Power Developed by the Net Force at the Final Speed

Power ($𝑃$) is the rate at which work is done. At the final speed, the power developed is given by:

$$𝑃=𝐹\cdot 𝑣$$

Given:

• Force, $𝐹=2343.75$ N
• Final velocity, $𝑣=25\text{ m/s}$

$$𝑃=2343.75\text{ N}\times 25\text{ m/s}=58593.75\text{ W}$$

So, the power developed by the net force at the final speed is $58593.75\text{ W}$ (Watts).

Summary:

• Acceleration of the car: $1.5625{\text{ m/s}}^2$
• Net force acting on the car: $2343.75\text{ N}$
• Work done by the net force: $468750\text{ J}$
• Power developed by the net force at the final speed: $58593.75\text{ W}$