A projectile is launched from ground level to the top of a cliff which is 195 m away and 135 m high.If the projectile lands on top of the cliff 6.5 s after it is fired, find the initial velocity of the projectile ( (a)magnitude and (b)direction ). Neglect air resistance.
To find the initial velocity of the projectile, we can use the equations of motion. First, let's break down the given information:
Distance (horizontal range) traveled by the projectile: 195 m
Height of the cliff: 135 m
Time taken for the projectile to reach the top of the cliff: 6.5 s
Neglecting air resistance, we can assume that the only forces acting on the projectile are gravity in the vertical direction and the initial velocity in the horizontal direction.
(a) Finding the initial velocity's magnitude:
To find the initial velocity's magnitude, we need to first determine the time it takes for the projectile to reach the maximum height. Since the projectile lands on top of the cliff, we can assume that the time taken to reach the top is half of the total time of flight:
Time for projectile to reach the top = 6.5 s / 2 = 3.25 s
Using this time and the given vertical displacement, we can find the initial vertical velocity (Vy) using the equation:
Vy = (final displacement - initial displacement) / time
Vy = (0 - 135 m) / 3.25 s
Vy ≈ -41.54 m/s
Next, we can use the equation of motion to find the initial vertical velocity (Vy) in terms of the initial velocity's magnitude (v0) and the angle of projection (θ):
Vy = v0 * sin(θ)
Solving for v0:
v0 = Vy / sin(θ)
v0 = -41.54 m/s / sin(θ)
(b) Finding the initial velocity's direction:
To find the direction of the initial velocity, we need to find the angle of projection (θ). We can use the horizontal range and the total time of flight to calculate it.
Using the equation for the horizontal range (R):
R = v0 * cos(θ) * t
195 m = v0 * cos(θ) * 6.5 s
Solving for cos(θ):
cos(θ) = 195 m / (v0 * 6.5 s)
To get the angle of projection:
θ = arccos(195 m / (v0 * 6.5 s))
Now that we have obtained the magnitude (v0) and the direction (θ) of the initial velocity, we can specify the initial velocity of the projectile.