A metal sphere is launched with an initial velocity of 1.5 m/s as it leaves the ramp. The end of the ramp is 1.20 m above the floor. Calculate the range of the sphere. (Range is the horizontal displacement of the projectile.)
If it leaves the ramp horizontally, if falls 1.2 m
Time to fall: 1.2=1/2 9.8 t^2
solve for time t.
then horizontal distance:
distance=1.5m/s*time t
Oops
To calculate the range of the sphere, we need to determine the time it takes for the sphere to hit the ground.
We can start by finding the time it takes for the sphere to reach its highest point using the equation:
v = u + at
where:
- v is the final velocity (0 m/s at the highest point)
- u is the initial velocity (1.5 m/s)
- a is the acceleration due to gravity (-9.8 m/s^2)
- t is the time taken
Rearranging the equation to solve for t:
0 = 1.5 - 9.8t
9.8t = 1.5
t ≈ 0.15 seconds
Next, we can calculate the time it takes for the sphere to fall from its highest point to the ground. Since the sphere falls from rest, we use the equation:
s = ut + 1/2at^2
where:
- s is the distance (1.20 m)
- u is the initial velocity (0 m/s)
- a is the acceleration due to gravity (-9.8 m/s^2)
- t is the time taken
Rearranging the equation to solve for t:
1.20 = 0*t + 1/2*(-9.8)*t^2
4.9t^2 = 1.20
t^2 ≈ 0.4898
t ≈ 0.7 seconds
Now, we can calculate the range (horizontal displacement) of the sphere using the equation:
range = initial velocity * time
range = 1.5 m/s * 0.7 s
range ≈ 1.05 meters
Therefore, the range of the sphere is approximately 1.05 meters.
To calculate the range of the sphere, we can use the kinematic equations of motion. The horizontal motion of the sphere is not affected by gravity, so we only need to consider its initial horizontal velocity.
The range of the sphere can be calculated using the formula:
Range = Horizontal velocity * Time of flight
To find the time of flight, we need to calculate how long it takes for the sphere to reach the floor from the end of the ramp. We can use the vertical motion equation:
Vertical displacement = Initial vertical velocity * Time + (1/2) * Acceleration * Time^2
In this case, the vertical displacement is 1.20 m (the height of the ramp), the initial vertical velocity is 0 m/s (the sphere is launched horizontally), and the acceleration is the acceleration due to gravity, which is approximately -9.8 m/s^2.
Plugging these values into the equation, we get:
1.20 = 0 * Time + (1/2) * (-9.8) * Time^2
Simplifying, we get:
1.20 = -4.9 * Time^2
Rearranging the equation, we have:
Time^2 = 1.20 / (-4.9)
Taking the square root of both sides, we get:
Time = √(1.20 / (-4.9))
Now that we have the time of flight, we can calculate the range using the initial horizontal velocity:
Range = 1.5 m/s * Time
Plugging in the value of Time, we can solve for the range.