**Derive the equation of motion for a
projectile moving under the influence of gravity**

A projectile is an object that is launched into the air and moves under the influence of gravity. The motion of a projectile is a classic example of a two-dimensional motion problem.

The
equation of motion for a projectile moving under the influence of gravity is
derived by considering the forces acting on the projectile and applying
Newton's Second Law.

**Derive the equation of motion for a projectile moving under the influence of gravity-**Consider a projectile of mass m
that is launched into the air with an initial velocity v0 at an angle θ above
the horizontal.

The projectile moves under the influence of gravity, which acts
downward with a force of mg, where g is the acceleration due to gravity. The
motion of the projectile can be described by its position vector r(t), which
gives the position of the projectile at time t.

The forces acting on the projectile are the gravitational force mg and the air resistance force, which we assume to be negligible. The gravitational force acts downward and can be resolved into two components: one parallel to the horizontal direction and one parallel to the vertical direction.

**Derive the equation of motion for a projectile moving under the influence of gravity-**The component parallel to the horizontal direction does
not affect the motion of the projectile in the vertical direction and can be
ignored. The component parallel to the vertical direction is given by:

- Fy = -mg

where the negative sign indicates
that the force is acting downward.

The equation of motion for the
projectile can be derived by applying Newton's Second Law, which states that
the net force acting on an object is equal to its mass times its acceleration.
In the case of the projectile, the net force acting on it is the gravitational
force, and its acceleration is given by the second derivative of its position
vector with respect to time:

- Fnet = ma = m(d^2r/dt^2)

where a is the acceleration of the
projectile.

Substituting the gravitational
force for the net force, we have:

- -mg = m(d^2r/dt^2)

Dividing both sides by m, we
obtain:

- -g = d^2r/dt^2

This is the equation of motion for
the projectile under the influence of gravity. It is a second-order
differential equation that describes the vertical motion of the projectile. The
solution to this equation gives the vertical position of the projectile as a
function of time.

To solve this equation, we need to
specify the initial conditions of the projectile, namely its initial position
and velocity. Let the initial position of the projectile be r0 = (x0, y0),
where x0 is the horizontal distance from the origin and y0 is the initial
height above the ground. Let the initial velocity of the projectile be v0 =
(v0x, v0y), where v0x is the horizontal component of the velocity and v0y is
the vertical component of the velocity.

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The initial conditions can be used
to determine the constants of integration in the solution to the equation of
motion. Integrating the equation with respect to time gives:

- -dy/dt = gt + vy0

Integrating again with respect to
time gives:

- y = y0 + vy0t - 1/2gt^2

**Derive the equation of motion for a projectile moving under the influence of gravity-**This is the equation of motion for
the vertical position of the projectile as a function of time. It gives the
height of the projectile above the ground at any time t.

Similarly, we can derive the
equation of motion for the horizontal position of the projectile. The
horizontal motion of the projectile is not affected by gravity, so the net
force acting on it is zero. The equation of motion for the horizontal position
of the projectile is:

- dx/dt = vx0

where vx0 is the horizontal component
of the initial velocity.

The equations of motion for the
vertical and horizontal positions of the projectile can be combined to give the
trajectory of the projectile. The trajectory is given by the parametric
equations:

- x = x0 + vx0t
- y = y0 + vy0t - 1/2gt^2

**Derive the equation of motion for a projectile moving under the influence of gravity-**These equations give the horizontal
and vertical positions of the projectile as functions of time. The trajectory
of the projectile is the path traced out by these positions as the projectile
moves through the air.

**Conclusion**

The equation of motion for a projectile moving under the influence of gravity is derived by considering the forces acting on the projectile and applying Newton's Second Law.

**Derive the equation of motion for a projectile moving under the influence of gravity-**The equation
of motion is a second-order differential equation that describes the vertical
motion of the projectile. The solution to this equation gives the vertical
position of the projectile as a function of time.

**Derive the equation of motion for a projectile moving under the influence of gravity-**The horizontal motion of the
projectile is not affected by gravity and can be described by a simple linear
equation. The trajectory of the projectile is given by the parametric equations
that combine the equations of motion for the vertical and horizontal positions
of the projectile.

**FAQ.**

**Q. What is the equation of motion for a projectile under
gravity?**

Ans. The equation of motion for a
projectile moving under the influence of gravity is given by: -g = d^2r/dt^2,
where g is the acceleration due to gravity and r(t) is the position vector of
the projectile at time t.

**Q. What does the equation of motion describe?**

Ans. The equation of motion
describes the vertical motion of the projectile, specifically the second
derivative of its position vector with respect to time.

**Q. How is the equation of motion derived?**

Ans. The equation of motion is
derived by considering the forces acting on the projectile, namely gravity, and
applying Newton's Second Law.

**Q. What are the initial conditions needed to solve the equation
of motion?**

Ans. The initial conditions needed
to solve the equation of motion are the initial position and velocity of the
projectile, which can be used to determine the constants of integration in the
solution.

**Q. How accurate is the equation of motion in real-world
situations?**

Ans. The equation of motion assumes
certain ideal conditions, such as negligible air resistance and a constant
acceleration due to gravity. In real-world situations, these conditions may not
hold, and the motion of the projectile may be more complex.

**Q. Can the equation of motion be used for non-vertical motion?**

Ans. The equation of motion derived
is specifically for the vertical motion of the projectile. For non-vertical
motion, additional equations and considerations may be needed to describe the
motion accurately.

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