4. A net force acts on an object initially at rest, giving it an acceleration, a, while moving it a distance, Ad, forward across a horizontal surface.

Question

T/l
(a) Use kinematics equations to show that v? = 2ad.
mv?
(b) The final total mechanical energy is Ek =
2 •
Substitute the equation for the final velocity into the equation for the final kinetic energy and simplify. What does this new equation prove?

Answer 1

(a)
Using kinematics equations, we can relate the initial velocity, final velocity, acceleration, and distance using the equation:
v^2 = u^2 + 2as
Where:
v = final velocity
u = initial velocity (0 when the object is initially at rest)
a = acceleration
s = distance

Since the object is initially at rest, u = 0. So the equation becomes:
v^2 = 0 + 2as
v^2 = 2as
v^2 = 2ad

So, v^2 = 2ad

(b)
The final kinetic energy of the object, Ek, is given by the equation:
Ek = 1/2 * m * v^2
Substitute the equation for final velocity into the equation for Ek:
Ek = 1/2 * m * (2ad)
Ek = m * ad

This new equation proves that the final kinetic energy is directly proportional to both the mass and acceleration of the object. It shows that the final kinetic energy is equal to the product of the mass and the distance moved multiplied by acceleration.