An soldier moves from (3,2) along positive X axis at speed of 4 meter per second. An archer is standing at (-3,9). The archer needs to shoot the soldier down. Also the archer knows that his arrow will travel at speed of 20 meters per second. Calculate the angle with X-axis, such that the arrow shot at that angle will kill the soldier.
the problem here is time at which they will hit is not given. So how can we calculate the angle? method of solving is important regardless the final answer
To make it easier, I shifted everything right by 3 units and down by 2 units. This makes the archer's initial x position 0 and y position 7, and the soldier's x position 6 and y position 0. You know that the two (arrow and soldier) x-positions will be equal when they meet. We know that the x-component of the arrow will be the initial x-component of the arrow's velocity multiplied by the time. Representing this as an equation we get Xarrow = 20cos(theta)*t where theta is the initial angle the arrow is fired and t is the time since it was fired. We also can find the y equation of the arrow. We will get Yarrow = 7 + 20sin(theta)t - 1/2 gt^2. Now for the soldier. The y-component is constantly 0 now that we shifted them down by 2 units. So, the equation of motion for the soldier is only in the x direction. We get Xsoldier = 4t + 6. Setting the x equation of the soldier equal to the one of the arrow solves t in terms of theta. You can plug this into the Yarrow equation to solve for theta. We know to set y = 7 because that will solve for the angle when the arrow was fired.