To determine whether the grenade will explode before hitting the ground, we need to calculate the time it takes for the grenade to reach the ground and compare it with the fuse time of 3 seconds.
First, we'll break down the initial velocity of the grenade into its horizontal and vertical components:
Initial velocity (v) = 7.50 m/s
Angle (θ) = 35 degrees below the horizontal
Horizontal component of velocity (v_x) = v * cos(θ)
Vertical component of velocity (v_y) = v * sin(θ)
Let's calculate the horizontal and vertical components of the initial velocity:
v_x = 7.50 m/s * cos(35°) ≈ 6.12 m/s
v_y = 7.50 m/s * sin(35°) ≈ 4.27 m/s
Now, we'll use the vertical motion equation to calculate the time (t) it takes for the grenade to reach the ground:
h = v_y * t - 0.5 * g * t^2
where:
h = height of the building = 17 m
g = acceleration due to gravity ≈ 9.81 m/s²
Plugging in the values and solving for t:
17 m = 4.27 m/s * t - 0.5 * 9.81 m/s² * t^2
0.5 * 9.81 m/s² * t^2 - 4.27 m/s * t + 17 m = 0
Solving this quadratic equation for t using the quadratic formula:
t = (-b ± sqrt(b² - 4ac)) / 2a
where:
a = 0.5 * 9.81 m/s² = 4.905 m/s²
b = -4.27 m/s
c = 17 m
Calculating the discriminant (b² - 4ac):
Discriminant = (-4.27 m/s)² - 4 * 4.905 m/s² * 17 m ≈ 11.148
Since the discriminant is positive, there are two real solutions for t. However, we are interested in the positive root (time can't be negative):
t = (-(-4.27) + sqrt(11.148)) / (2 * 4.905) ≈ 1.197 seconds
The time it takes for the grenade to reach the ground is approximately 1.197 seconds, which is less than the 3-second fuse time. Therefore, the grenade will explode before hitting the ground.