Two projectiles are launched from a cliff of height H = 16m as shown in the diag

Question

Two projectiles are launched from a cliff of height H = 16m as shown in the diagram below. Projectile 1 is launched with velocity v1 = 12 m/s at an angle q1, where q1 = 35 degrees. Projectile 2 is launched with velocity v2 = 12 m/s at an angle q2, where q2 = 50 degrees. The next two questions pertain to this physical situation.

a)  If the projectiles are lauched at the same time, which projectile lands first?

Projectile 1 lands first.

Projectile 2 lands first.

The projectiles land at the same time.

Here we have two vectors V and W. The angle between these vectors is A.

b)  What is the component of V parallel to W in terms of A and the magnitudes of V and W?

V

W

V sin(A)

V cos(A)

W sin(A)

W cos(A)

c)  What is the component of V perpendicular to W in terms of A and the magnitudes of V and W?

V

W

V sin(A)

V cos(A)

W sin(A)

W cos(A)

Now assume A=(3 m, 4 m) and B=(5 m, -3 m), where the first number is the x-component of the vector, and the second number is the y-component of the vector.

d)  What is the magnitude of vector A?

7 m

4 m

(8 m, 1 m)

5 m

(3 m, 4 m)

e)  What is A + B?

(8 m, 1 m)

7 m

(3 m, 1 m)

10.83 m

5.83 m

(7 m, 2 m)

Explanation / Answer

a) We need to find out the component of velocity of the projectiles in the vertical direction to figure out which will land earlier.

Beacuse the acceleration (due to gravity) and the distance to be travelled (height H) is the same for both cases, the projectile with higher intial intial velocity will reach earlier.

Vertical component for 1: v1*sin(q1) = 12*sin(35) =6.88 m/sec

Vertical component for 1: v1*sin(q1) = 12*sin(50) = 9.12 m/sec

Now, these velocities are in the negative direction to the distance to be travelled.
Therefore, when taken with sign, 1 has a higher velocity. Thus, 1 will reach earlier.

b) V cos(a) : Dividing the vector into two components such that one is parallel to W and one is perpendicular, we get that component of V parallel to W is V Cos(A).

c) V Sin(A). based on similar logic as (b)

d) The magnitude of any vector is the lenght of that vector if plotted on the X-Y cordinate axis. Therfore, in this case, on plotting point A, we get that Lenght of A is sqrt(32+42) which is 5m.

e) The sum of two vectors should be a vector. Therefore, A+B will be a vector.
In a vector, the two components are independant of each other. Therefore, we will sum X components together and Y components together.
X components: 3+5 =8m.

Y components: 4 +(-3) = 1m.

Therfore the answer is (8m,1m)

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