Question #b606a
1 Answer
Their launch velocities must be in a ratio of
Explanation:
As you know, you can break down the trajectory of an object launched at an angle
This means that you can do the same for its launch velocity,
#v_(0x) = v_0 * costheta -># horizontal component
and
#v_(0y) = v_0 * sintheta -># vertical component
Now, you know that the maximum heights of the two objects must be equal. You can focus solely on the vertical component of the movement, which is influenced by the gravitational acceleration,
At maximum height, the vertical component of the object's velocity will be equal to zero. This means that you can write
#overbrace(v_"top on y"^2)^(color(blue)(=0)) = v_(01y)^2 - 2 * g * h_1 -># for object 1
and
#overbrace(v_"top on y"^2)^(color(blue)(=0)) = v_(02y)^2 - 2 * g * h_2 -># for object 2
You know that
#v_(01y)^2 = 2 * g * h#
#h = v_(01y)^2/(2 * g) = [v_(01) * sin(theta_1)]^2/(2 * g)#
#h = (v_(01)^2 * [sin(30^@)]^2)/(2 * g) = [v_(01)^2 * (1/2)^2]/(2g) = 1/8 * v_(01)^2/g#
For the second object, you have
#h = v_(02y)^2/(2 * g) = [v_(02) * sin(theta_2)]^2/(2 * g)#
#h = (v_(02)^2 * [sin(60^@)]^2)/(2 * g) = [v_(01)^2 * (sqrt(3)/2)^2]/(2g) = 3/8 * v_(02)^2/g#
Here
The ratio between these two initial velocities will be
#1/color(red)(cancel(color(black)(8))) * v_(01)^2/color(red)(cancel(color(black)(g))) = 3/color(red)(cancel(color(black)(8))) * v_(02)^2/color(red)(cancel(color(black)(g)))#
#v_(01)^2/v_(02)^2 = 3 implies v_(01)/v_(02) = color(green)(sqrt(3))#
Notice that this is the ratio between the values of
#h = [v_(01)^2 * sin^2(theta_1)]/(2 * g) implies v_(01)^2 = (2 * g * h)/(sin^2theta_1)#
Similarly, you have
#v_(02)^2 = (2 * g * h)/(sin^2theta_2)#
Divide these expressions to get
#v_(01)^2/v_(02)^2 = color(red)(cancel(color(black)(2 * g * h)))/sin^2theta_1 * sin^2theta_2/color(red)(cancel(color(black)(2 * g * h))) = sin^2theta_2/sin^2theta_1#
This is equivalent to
#v_(01)/v_(02) = sqrt(sin^2theta_2/sin^2theta_1) = sintheta_2/(sintheta_1) = sqrt(3)/color(red)(cancel(color(black)(2))) * color(red)(cancel(color(black)(2)))/1 = color(green)(sqrt(3))#