# [Physics] Kinematics and Energy – Finding the work done in projectile motion

homework-and-exercisesnewtonian-gravitynewtonian-mechanicsprojectilework

If you are asked to find the work done by gravity on a projectile, the force will obviously be the mass multiplied by gravitational acceleration. What will the displacement be, the horizontal or vertical one?

The work is given by:

$$W = \int \vec{F}.d\vec{r}$$

where the force $\vec{F}$ and the displacement $d\vec{r}$ are both vectors and $.$ is the dot product. We can write $d\vec{r}$ as:

$$d\vec{r} = d\vec{x} + d\vec{y}$$

where $d\vec{x}$ and $d\vec{y}$ are the displacements in the horizontal and vertical directions respectively, and substituting this into our expression for the work we get:

$$W = \int \vec{F}.(d\vec{x} + d\vec{y}) = \int \vec{F}.d\vec{x} + \int \vec{F}.d\vec{y}$$

But the force $\vec{F}$ is directed downwards, i.e. in the $\vec{y}$ direction, so:

$$\vec{F}.d\vec{x} = 0$$

and:

$$\vec{F}.d\vec{y} = Fdy$$

so:

$$W = \int Fdy$$

and since $F$ is a constant we can take it out of the integral to get:

$$W = F \int dy = F\Delta y$$

where $\Delta y$ is the $y$ distance moved. So the work is the force due to gravity times the vertical distance moved.