Example: Finding the range of a projectile

In the previous example of time of flight, the out of control bus is forced to jump from a location $\langle 0,40, - 5\rangle$m with an initial velocity of $\langle 80,7, - 5\rangle m/s^{- 1}$. We have now found the time of flight to be 3.65s and now want to find the position of where the bus returns to the ground.

Facts

Starting position of the bus $\langle 0,40, - 5\rangle$
Initial velocity of the bus $\langle 80,7, - 5\rangle$
The acceleration due to gravity is 9.8 $\frac{m}{s^{2}}$ and is directed downward.
The bus experiences one force - the gravitational force (directly down).
The bus takes 3.65s to reach the ground (from previous problem)

Lacking

The final position of the bus.

Approximations & Assumptions

Assume no drag effects
Assume ground is when position of bus is 0 in the y direction

Representations

Diagram of forces acting on bus once it leaves the road.

The general equation for calculating the final position of an object:

$$ {\overset{\rightarrow}{r}}_{f} = {\overset{\rightarrow}{r}}_{i} + {\overset{\rightarrow}{v}}_{avg}\Delta t $$

Also know as the position update formula.

Solution

From the previous problem you already know the final location of the ball in the y direction to be 0 as it has met the ground after 9.59s.

We now to find the range in the x and z directions in order to have a position vector for the final resting place of the bus.

There is no force acting in the x or z directions as the only force acting on the system is the gravitational force which acts in the y-direction.

This means that the initial velocities in both of these directions have remained unchanged.

We know the amount of time the bus has been traveling in the x-direction at its initial velocity and its initial position so we can compute the distance travelled in this direction using the position update formula for x-components.

$$ x_{f} = x_{i} + V_{avg,x}\Delta t $$

Plug in respective values for variables.

$$ = 0 + 80m/s(3.65s) $$

Compute range in x-direction.

$$ = 292m $$

Repeat same process for the z-components:

$$ z_{f} = z_{i} + V_{avg,z}\Delta t $$

Plug in respective values for variables.

$$ = - 5 + - 5m/s(3.65s) $$

Compute range in z-direction.

$$ = - 23.25m $$

Write range(final position vector) using all components:

Final position =

$$ \langle 292,0, - 23.255\rangle m $$