Question

. Show that 90 J of work is needed to increase the speed of
a 20-kg cart by 3 m/s.

Answer 1

Answer on Question #39463, Physics, Mechanics | Kinematics | Dynamics

Show that 90 J of work is needed to increase the speed of a 20-kg cart by 3 m/s.

**Solution:**

In order to accomplish work on an object there must be a force exerted on the object and it must move in the direction of the force.

Work = Force x distance moved in direction of force

$$W = F \cdot d$$

Newton's Second Law

$$F = m a$$

Kinematics equation

$$2 a d = v ^ {2} - v _ {0} ^ {2}$$

where $a$ is acceleration, $d$ is distance, $v_{0}$ is initial velocity and $v$ is final velocity.

Thus,

$$W = F \cdot d = m a d = \frac {m}{2} (v ^ {2} - v _ {0} ^ {2})$$

When $v_{0} = 0, v - v_{0} = \Delta v = 3 \, \text{m/s}$

$$v ^ {2} - v _ {0} ^ {2} = (v + v _ {0}) (v - v _ {0}) = \Delta v \cdot \Delta v = (\Delta v) ^ {2}$$

Thus,

$$W = \frac {m}{2} (v ^ {2} - v _ {0} ^ {2}) = \frac {m (\Delta v) ^ {2}}{2} = \frac {2 0 \cdot (3) ^ {2}}{2} = 9 0 \, \text{J}$$